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ARITHMETIC 


-* 


j&m 


ARITHMETIC 


FOR 


SCHOOLS 


BY 


CHARLES   SMITH,   M.A. 
it 

MASTER  OF  SIDNEY   SUSSEX  COLLEGE,   CAMBRIDGE 
REWRITTEN   AND   REVISED   BY 

CHARLES   L.   HARRINGTON 

HEAD  MASTER  OF  DR.  J.  SACHS'S  SCHOOL 
FOR  BOYS,  NEW  YORK 


MACMILLAN    AND    CO. 

AND      LONDON 
1895 

All  rights  reserved 


Copyright,  1895, 
By  MACMILLAN  AND  CO. 


NortoaoD  -press : 

J.  S.  dishing  &  Co.  —  Berwick  &  Smith. 

Norwood,  Mass.,  U.SA 


PREFACE. 


In  the  following  work  it  has  been  the  endeavor  to  put 
the  science  of  Arithmetic  on  a  sound  basis,  and  to  give 
clear  and  complete  explanations  of  all  the  fundamental 
principles  und  processes.  It  has  not  been  the  aim  to 
introduce  novelties,  but  to  promote  accuracy  and  clearness 
of  conception,  so  as  to  make  the  study  of  Arithmetic  not 
only  of  practical  utility,  but  also  of  great  educational  value. 

I  am  indebted  to  many  friends  for  their  kindness  in 
looking  over  the  proof  sheets,  for  help  in  the  verification 
of  the  answers,  and  for  valuable  criticisms  and  sugges- 
tions. My  special  thanks  are  due  to  Mr.  J.  Barnard,  M.A., 
Head  Mathematical  Master  at  Christ's  Hospital. 

CHARLES  SMITH. 


184003 


CONTENTS. 


Chapter 

I.    Numeration  —  Notation 

II.    Addition  —  Subtraction 

ION      .... 


Multiplication  —  Drvrs- 


III.   Factors  and  Multiples — Square  Boot  —  Highest 
Common    Factor  —  Least    Common    Multiple  — 


Page 

1 

13 


Parenthesis  —  Canc 

ELLATION 

.      59 

IV.   Fractions     .... 

87 

V.    Decimal  Measures     . 

125 

VI.   Non-Decimal  Measures 

139 

VII.    Approximation     . 

176 

/"III.    Areas  —  Volumes 

185 

Carpeting,  Papering,  Plastering 

189 

Dimensions  of  Circles 

193 

Specific  Gravity 

201 

IX.    Ratio  and  Proportio 

v 

208 

Partnership. 

217 

Mixtures    . 

221 

Work  and  Time 

223 

Races  and  Games 

226 

X.    Percentages 

231 

Profit  and  Loss . 

234 

Trade  Discount 

. 

237 

Commission  and  Brokerage 

239 

Taxes  and  Duties 

241 

ix 

X 


CONTENTS. 


Chapter  Page 

XL    Interest 244 

Promissory  Notes 244 

Interest  for  Days  at  6  % 249 

Interest  for  Days  at  Other  Rates  than  6  %         .         .  252 

6  %  for  Long  Times 254 

Annual  Interest 256 

Commercial  Discount        .         .    -     .        .         .        .  258 

Exact  Interest 268 

Partial  Payments 269 

XII.    Exchange — Domestic  and  Foreign          .        .         .  275 

XIII.  Stocks  and  Bonds 282 

XIV.  Progressions 291 

XV.   Cube  Root 297 


OF  THE 

UNIVERSITY 

OF 


ARITHMETIC. 

CHAPTER   I. 

NUMEKATION  —  NOTATION. 

1.  The  idea  of  number  is  first  acquired  from  the  obser- 
vation of  groups  of  distinct  objects,  actions,  sounds,  etc. ; 
we  thus  learn  to  speak  of  two  boys,  three  balls,  four 
strokes  of  a  clock,  etc. 

A  single  object  of  any  kind,  or  any  group  of  objects 
considered  as  a  whole,  is  called  a  unit. 

Thus,  one  ball,  one  inch,  one  dozen,  one,  one  ten,  are  units. 

2.  Arithmetic  is  the  science  which  treats  of  numbers  and 
of  the  different  operations  to  which  they  are  subject. 

3.  The  first  few  numbers  in  order  are,  one,  two,  three, 
four,  Jive,  six,  seven,  eight,  nine,  and  ten. 

4.  It  will  be  observed  that  the  names  of  the  first  ten 
numbers  are  in  no  way  connected  with  one  another. 

Now  it  is  obvious  that  the  knowledge  of  numbers  and 
of  their  relations  to  one  another  must  always  have  re- 
mained very  limited  if  every  successive  number  had  had 
a  special  name  given  to  it  independent  of  the  names  of 
the  preceding  numbers ;  for  it  would  be  almost  impossi- 
ble to  remember,  in  their  order,  many  such  names. 
b  1 


2  NUMERATION  —  NOTATION.  [Chap.  I. 

5.  Successive  numbers  have  therefore  been  named 
according  to  a  systematic  plan  which  requires  the  use  of 
as  few  independent  names  as  possible. 

The  method  by  which  numbers  are  expressed  in  words 
according  to  some  systematic  plan  is  called  Numeration. 


Numeration. 

6.  To  show  how  all  numbers  can  be  named  by  means 
of  a  few  special  words,  imagine  a  collection  of  objects  of 
the  same  kind,  for  example,  a  heap  of  apples ;  and  sup- 
pose that  we  wish  to  know  how  many  apples  there  are, 
and  to  give  a  name  to  this  number. 

If  there  are  not  more  than  ten  apples  altogether,  we 
find  the  number  at  once  by  counting  them,  that  is  by  say- 
ing in  order  the  names  one,  two,  three,  etc.,  each  time 
separating  one  of  the  apples  from  the  original  heap ;  and 
the  name  which  is  said  with  the  last  of  the  heap  gives 
the  number  of  the  apples. 

If  there  are  more  than  ten  apples  in  the  heap,  count 
off  ten  and  put  them  apart,  and  go  on  making  groups  of 
ten  until  there  are  fewer  than  ten  apples  left.  Suppose 
there  are  seven  groups  of  ten  each  and  five  apples  over, 
then  we  could  call  the  number  seven  tens  and  five. 

By  separating  the  whole  heap  into  groups  of  ten  in 
this  way  we  at  once  find,  and  can  give  a  name  to,  the 
number  of  the  apples,  provided  there  are  not  more  than 
ten  of  the  groups.  Thus,  our  original  ten  names  suffice 
to  name  all  numbers  up  to  that  which  is  made  up  of  ten 
groups,  each  containing  ten  apples,  and  we  have  a  new 
name,  namely  one  hundred,  for  the  number  which  con- 
sists of  ten  tens. 

If  there  are  more  than  ten  of  the  groups  each  of  which 
contains  ten  apples,  the  groups  can  be  arranged  in  sets  of 


Arts.  5-7.]  NUMERATION.  3 

ten,  so  that  there  will  be  one  hundred  apples  in  each  of 
these  sets.  Suppose  that  there  are  five  of  these  sets  and 
six  groups  over  and  four  single  apples  besides,  then  the 
number  is  made  up  of  five  hundreds,  six  tens,  and  four. 
Thus  no  new  name  is  necessary  until  we  come  to  the 
number  which  consists  of  ten  hundreds,  and  this  number 
is  called  a  thousand. 

It  will  be  seen  at  once  that  the  names  in  actual  use  are  only 
slightly  modified  forms  of  the  names  which  naturally  arise  from 
the  above  method  of  division  into  groups  of  ten.  Instead  of 
saying  two  tens,  three  tens,  four  tens,  etc.,  we  say  twenty,  thirty, 
forty,  etc.,  and  we  say  seventy -five  instead  of  seventy  and  five. 
Also  instead  of  the  names  ten  and  one,  ten  and  two,  ten  and 
three,  ten  and  four,  etc.,  we  use  the  names  eleven  [Gothic  ainlif, 
ain  one  and  lif  ten],  twelve  [Gothic  twalif,  twa  two  and  lif  ten], 
thirteen,  fourteen,  etc. 

If  an  apple  be  cut  into  ten  equal  parts,  any  number  of 
these  parts  may  be  put  with  some  apples  already  counted. 
Each  part  is  but  one  out  of  ten  parts  and  may  be  counted 
as  one  tenth. 

If  one  of  these  tenths  be  cut  into  ten  equal  parts,  each 
new  part  is  but  one  out  of  ten  parts  of  one  tenth,  and 
may  be  counted  as  one  one-hundredth.  Thus  by  separat- 
ing into  ten  equal  parts,  etc.,  we  do  not  really  require  new 
names.  If,  now,  we  have  five  sets,  six  groups,  four  single 
apples,  and  three  tenths  and  seven  hundredths,  then  the 
number  is  five  hundred  sixty-four  and  thirty-seven  hun- 
dredths (for  three  tenths  is  equal  to  thirty  hundredths). 

The  principle  of  the  ordinary  system  of  numeration 
will  now  be  apparent. 

7.  The  English  names  which  are  employed  in  the 
system  of  Numeration  which  is  universally  used  are  the 
following:  one,  two,  three,  four,  Jive,  six,  seven,  eight,  nine, 
ten,  eleven,  twelve,  thirteen,  fourteen,  fifteen,  sixteen,  seven- 


4  NUMERATION  — NOTATION.  [Chap.  I. 

teen,  eighteen,  nineteen,  twenty,  thirty,  forty,  fifty,  sixty, 
seventy,  eighty,  ninety, 


a  hundred 

which 

is     ten  tens, 

a  thousand 

u 

"     ten  hundreds, 

a  million 

a 

"     a  thousand  thousands, 

a  billion 

a 

"     a  thousand  millions, 

a  trillion 

u 

"     a  thousand  billions, 

a  quadrillion 

a 

"     a  thousand  trillions, 

and  so  on. 

The  names  billion,  trillion,  etc.,  are  very  rarely  used. 

8.  The  numbers  one,  ten,  a  hundred,  a  thousand,  ten 
thousand,  a  hundred  thousand,  a  million,  etc.,  are  often 
called  units  of  the  first  order,  of  the  second  order,  of  the 
third  order,  etc. ;  and  ten  units  of  any  order  are  required 
to  make  one  unit  of  the  next  higher  order. 

To  give  a  name  to  any  number  it  is  sufficient  to  state 
the  number  of  units  of  each  different  order  that  the 
number  contains. 

Thus  the  number  which  is  made  up  of  three  millions  two  hun- 
dreds-of-thousands  three  tens-of-thousands  four  thousands  five  hun- 
dreds seven  tens  six  units  two  tenths  and  four  hundredths  is  called 
three  million  two  hundred  thirty-four  thousand  five  hundred 
seventy-six  and  twenty-four  hundredths ;  also  the  number  which  is 
made  up  of  two  hundreds-of-millions  three  tens-of-millions  two 
tens-of-thousands  three  thousands  and  five  hundreds  is  called  two 
hundred  thirty  million  twenty-three  thousand  five  hundred. 

It  should  be  remarked  that  the  parts  of  a  number  are 
mentioned  in  the  order  of  their  magnitude,  the  largest 
being  given  first;  in  English,  however,  this  order  is 
reversed  for  the  numbers  between  ten  and  twenty.  It 
should  also  be  noticed  that  all  the  thousands,  all  the  mil- 
lions, all  the  billions,  etc.,  are  grouped  together,  as  in  the 
above  two  cases. 


Arts.  8-11.]  NOTATION.  5 

9.  The  system  of  numeration  described  above  is  called 
the  Decimal  system,  since  ten  units  of  any  order  are 
required  to  make  up  one  unit  of  the  next  higher  order. 

The  decimal  system  of  numeration  is  employed  by  all 
people  who  have  any  names  at  all  for  numbers  greater 
than  ten,  and  the  origin  of  the  system  was  doubtless  the 
natural  habit  of  counting  on  the  fingers. 

Notation. 

10.  We  have  now  to  show  how  numbers  can  be  repre- 
sented in  a  simple  manner  by  means  of  a  few  symbols 
called  figures  or  digits. 

The  method  by  which  numbers  are  expressed  by  means 
of  symbols  according  to  some  systematic  plan  is  called 
Notation. 

11.  The  first  nine  numbers  in  order  are  represented 
by  the  symbols 

1,  2,  3,  4,  5,  6,  7,  8,  9. 

The  same  figures  are  also  employed  to  represent  the 
first  nine  collections  of  tens,  of  hundreds,  of  tJwusands, 
etc.,  but  on  the  understanding  that  the  figures  are  to  be 
written  in  a  row,  and  that  the  figure  which  represents 
the  units  of  the  highest  order  named  in  a  number  is  to 
be  written  as  the  left-hand  figure  of  the  row;  while 
the  figure  which  represents  the  units  of  the  lowest  order 
named  is  to  be  written  as  the  right-hand  figure.  Thus, 
any  figure  placed  just  to  the  right  of  another  represents 
units  of  the  order  next  below  that  represented  by  the 
other. 

To  distinguish  the  figures  which  represent  units,  tens, 
hundreds,  etc.,  from  those  which  represent  tenths,  hun- 
dredths, etc.,  a  dot,  called  the  Decimal  Point,  must  be 


6  NUMERATION  — NOTATION.  [Chap.  I. 

written  between  the  figures  which  represent  units  and 
tenths. 

Forty- five  is  written 45. 

Four  hundred  seventy-two  is  written 472. 

Three  and  two-tenths  is  written 8.2 

Fifty-one  and  twenty-seven  hundredths  is  written  51.27 

Thirty -five  hundredths  is  written .35 

12.  The  decimal  point  serves  to  separate  the  figures 
which  represent  the  wholes  from  those  which  represent 
the  tenths,  hundredths,  etc.  That  part  of  the  number 
to  the  left  of  the  decimal  point  is  called  the  Integral  part ; 
that  part  to  the  right  is  called  the  Decimal  part. 

In  2.5,  the  2  is  integral,  and  the  .5  is  decimal ;  in  45.627,  the  45 
is  integral,  and  the  .627  is  decimal. 

The  decimal  point  is  omitted,  if  the  number  contains 
no  decimal. 

13.  Besides  the  nine  symbols  already  specified,  it  is 
necessary  to  have  an  additional  symbol  to  meet  the  case 
when  units  of  one  or  more  of  the  different  orders  are 
absent.  This  symbol  is  0 ;  its  name  is  naught  or  cipher. 
It  has  no  value  by  itself,  and  is  used  to  indicate  that 
there  are  no  units  of  the  particular  order  corresponding 
to  the  place  in  which  it  occurs. 

The  other  figures  are  sometimes  distinguished  from 
the  naught  by  being  called  significant  figures. 

Thus,  20  represents  two  tens  and  no  ones;  that  is,  the  num- 
ber twenty.  Again,  2005  represents  two  thousands,  no  hundreds, 
no  tens,  and  five  ones ;  that  is,  the  number  two  thousand  five,  the 
naughts  serving  to  bring  the  significant  figures  into  the  places 
intended  for  them. 

It  should  be  noticed  that  a  naught  placed  at  the  beginning  of 
an  integral  number,  or  at  the  end  of  a  decimal  number,  does  not 
affect  the  value  of  the  number  •  056,  .7020,  0102,60  are  the  same 
as  56,  .702,  102.6. 


Arts.  12-15.]    .  NOTATION.  7 

14.  Any  figure,  say  5,  has  two  values ;  namely  (1)  its 
digit  value,  which  is  indicated  by  its  shape,  and  in  virtue 
of  which  it  always  represents  Jive  units  of  some  kind, 
and  (2)  its  local  value,  which  depends  only  on  its  place, 
and  in  virtue  of  which  it  represents  units  of  the  order 
which  corresponds  to  its  position  in  the  horizontal  row ; 
so  that  if  5  be  in  the  first  place  to  the  left  of  the  decimal 
point,  it  represents  five  units ;  if  in  the  second  place,  it 
represents  five  tens,  and  so  on ;  if  it  be  in  the  first  place 
to  the  right  of  the  decimal  point,  it  represents  5  tenths, 
and  so  on. 

Thus,  in  45  the  5  represents  five  ones;  in  567,  it  represents  five 
hundreds;  in  2.35,  it  represents  five  hundredths.  Again,  534.62 
represents  five  hundreds  three  tens  four  ones  six  tenths  and  two 
hundredths;  therefore  the  number  represented  is  Jive  hundred 
thirty-four  and  sixty-two  hundredths. 

15.  The  names  of  the  units  of  the  orders  in  common 
use,  and  the  positions  which  correspond  to  them,  are 
shown  below : 


I  s  I    J  s  §  §  1  £    I J I 


■5   E 


eo    E 


3d  2d 

Period  Period 


INTEGRAL  PERIODS  DECIMAL  PERIODS 

AND  ORDERS  AND  ORDERS 


It  will  be  seen  that  the  names  are  repeated  in  groups  of  three  ; 
and  that  the  decimal  periods  and  orders  correspond  in  name  to  the 
integral  periods  and  orders. 


8  NUMERATION  — NOTATION.      .         [Chap.  I. 

The  first  twelve  integral  periods  are  as  follows  : 

First,      Units.  Seventh,    Quintillions. 

Second,  Thousands.  Eighth,      Sextillions. 

Third,     Millions.  Ninth,       Septillions. 

Fourth,  Billions.  Tenth,       Octillions. 

Fifth,      Trillions.  Eleventh,  Nonillions. 

Sixth,      Quadrillions.  Twelfth,    Decillions. 

16.  To  write  in  figures  any  number  expressed  in  words, 

it  is  necessary  only  to  write  the  figures  which  represent 
the  number  of  the  units  of  the  different  orders  in  their 
proper  places  as  shown  above,  filling  up  the  vacant  places, 
if  any,  with  naughts. 

Thus,  to  write  in  figures  the  number  two  hundred  forty-three, 
we  must  have  2  in  the  place  for  hundreds,  4  in  the  place  for  tens, 
and  3  in  the  place  for  units.  To  write  in  figures  the  number  five 
hundred  twenty-four  thousand  six  hundred  seven,  we  must  have  5 
in  the  place  for  hundreds  of  thousands,  2  in  the  place  for  tens  of 
thousands,  4  in  the  place  for  units  of  thousands,  6  in  the  place  for 
hundreds,  0  in  the  place  for  tens,  and  7  in  the  place  for  units,  as 
follows  :  524,607.  Sixteen  million  is  written  16,000,000.  To  write 
in  figures  one  thousand  thirty  and  seven  thousandths,  we  must 
have  1  in  thousands'1  place,  0  in  hundreds'  place,  3  in  tens'1  place,  0 
in  units'  place,  0  in  tenths'1  place,  0  in  hundredths"  place,  and. 7  in 
thousandths'  place,  as  follows  :  1030.007. 

17.  To  express  in  words  any  number  given  in  figures, 
first  divide  the  integral  and  the  decimal  parts  separately 
into  groups  of  three,  beginning  at  the  right  in  each  case 
(the  left-hand  groups  will  often  be  incomplete)  ;  begin- 
ning at  the  left,  read  each  group  of  the  integral  part  as  if 
it  were  alone  and  give  it  the  name  of  the  period  to  which 
it  belongs,  then  read  the  decimal  part  as  if  it  were  inte- 
gral and  give  it  the  name  of  the  order  on  the  right. 

For  example,  to  express  in  words  the  number  represented  by 
24160523,  we  can  separate  off  two  complete  groups  of  three  figures, 
and  24,160,523  is  then  read  twenty-four  million  one  hundred 
sixty  thousand  five  hundred  twenty-three. 


Arts.  16,  17.]  EXAMPLES.  9 

To  express  in  words  the  number  represented  by  36405.4916, 
we  can  separate  off  one  complete  group  in  the  integral  part  and 
one  in  the  decimal  part,  and  36,405.4,916  is  then  read  thirty-six 
thousand  four  hundred  five  and  four  thousand  nine  hundred  six- 
teen ten-thousandths. 

To  'read  off'  decimals,  it  is,  however,  the  common 
practice  merely  to  name  the  digits  in  order. 

Thus  .615  is  read  '  decimal  six,  one,  five ' ;  15.0524  is  read  '  fifteen, 
decimal  naught,  five,  two,  four';  and  1567.0082  is  read  'one 
thousand  five  hundred  sixty-seven,  decimal  naught,  naught,  eight, 
two.' 

In  reading  numbers  the  word  '  and '  should  be  used  only  when 
we  reach  a  decimal  point. 

EXAMPLES  I. 

1.  For  what  does  5  stand  in  the  numbers  15, 1.57,  514, 
352167,  and  3561234,  respectively  ? 

2.  For  what  does  7  stand  in  the  numbers  70,  37123, 
125.479,  274126315,  and  370001002003,  respectively  ? 

3.  Name  all  the  figures  which  represent  their  digit 
value  of  hundred's  in  314,  2167,  50412,  and  31024. 

4.  Name  all  the  figures  which  represent  their  digit 
value  of  thousands  in  2314,  56123,  60417,  and  3005167. 

5.  Express  in  words  the  separate  value  of  every  figure 
in  3.5,  15.7,  125.34,  12.53,  800.17,  1200.63,  .875,  50.037, 
5.00107,  560002.19007. 

6.  Express  in  words  the  numbers  27,  349,  560,  3.06, 
1204,  and  5020. 

7.  Express  in  words  the  numbers  200.9,  6050,  12345, 
10.305,  40050,  and  1.20463. 

8.  Express  in  words  the  numbers  518618,  602010, 
100010,  504075,  420040,  and  107.005. 


10  NUMERATION  — NOTATION.  [Chap.  I. 

9.    Express  in  words  111111111,  1203405,  2314100, 
504.0314,  20050060,  and  30300074. 

10.  Express  in  words  the  numbers  3012004, 1101.11011, 
201201201,  1000040305101,  and  604102000300004. 

11.  "Write  in  figures  the  numbers  fifty-eight,  eighty- 
five,  two  hundred  eleven,  three  thousand  twelve,  six 
thousand  forty,  and  nine  thousand  three  hundred. 

Write  in  figures  the  following  numbers : 

12.  Twelve  and  three-tenths. 

13.  Three  hundred  four  and  nine-tenths. 

14.  Twenty-five,  three-tenths,  and  four  hundredths. 

15.  Eour,  six-tenths,  and  seven-thousandths. 

16.  One  million,  four-tenths,  and  three-millionths. 

17.  Write  in  figures  the  numbers  eleven  hundred 
eleven,  fourteen  hundred  sixty,  twelve  hundred  thou- 
sand sixteen  hundred,  six  million  twelve  hundred  six- 
teen, and  eleven  billion  eleven  hundred  eleven. 

18.  Write  in  figures  twenty  million  twenty  thousand, 
seventeen  million  fifty  thousand  nineteen,  one  hundred 
four  million  six  hundred  two  thousand  eleven,  and  six 
thousand  three  hundred  seven  million  two  thousand  fifty 
six. 

18.  The  ordinary  system  of  notation  was  introduced 
into  Europe  by  the  Arabians,  and  is  still  called  the  Ara- 
bic system  of  Notation  although  it  is  now  known  that  the 
Arabians  derived  their  knowledge  from  the  Hindoos. 

Eoman  Numerals. 

19.  Besides  the  Arabic  system  of  notation  some  use 
is  still  made  of  the  cumbrous  system  employed  by  the 
Romans. 


Arts.  18-22.]  ROMAN  NUMERALS.  H 

The  symbols  which  were  used  by  the  Romans,  and 
which  are  called  Roman  Numerals,  are  the  following : 

I  for  I,  V  for  5,  X  for  10,  L  for  50,  C  for  100,  D  for  500, 
M  for  1000. 

A  horizontal  line  over  any  numeral  increases  its  value 
one  thousand  fold  :  thus  V  stands  for  5000,  X  for  10000, 
etc. 

Roman  numerals  are  arranged  in  order  of  magnitude 
from  left  to  right,  and  are  repeated  as  often  as  may  be 
necessary. 

Thus,  2  is  represented  by  II,  30  by  XXX,  233  by  CCXXXIII,  and 
1887  by  MDCCCLXXXVII. 

20.  To  avoid  some  of  the  troublesome  repetitions 
which  are  common  to  the  Roman  system  of  notation,  a 
numeral  is  in  certain  cases  placed  before  another  of 
greater  value  to  denote  that  the  value  of  the  larger  is  to 
be  diminished  by  the  amount  of  the  smaller. 

Thus,  IV  denotes  one  less  than  five,  that  is,  4  ;  IX  denotes  one 
less  than  ten,  that  is,  9  ;  XL  denotes  ten  less  than  fifty,  that  is,  40  ; 
and  XC  denotes  ten  less  than  one  hundred,  that  is,  90 ;  so  also, 
CCXC  denotes  290. 

21.  The  symbols  CIO,  CCIOO,  CCCIOOO,  etc.,  were 
anciently  employed  to  denote  respectively  1000,  10,000, 
100,000  etc. ;  also  10,  IOO,  1000,  etc.,  to  denote  respect- 
ively 500,  5000,  50,000,  etc.  In  fact,  M  and  D  are  only 
modified  forms  of  CIO  and  10  respectively. 

22.  Roman  numerals  were  used  only  to  register  num- 
bers, and  were  never  employed  in  making  numerical 
calculations.  The  Romans  made  their  calculations  by 
means  of  counters  and  a  mechanical  apparatus  called  an 
Abacus.  The  counters  used  were  often  pebbles  (Latin, 
calculus),  whence  our  word  calculation. 


12  NUMERATION  — NOTATION.      [Chaps.  L,  II. 

EXAMPLES  II. 

1.  Express  all  the  numbers  from  1  to  20  by  means  of 
Eoman  numerals. 

2.  Express  by  means  of  Eoman  numerals  the  num- 
bers 20,  30,  40,  50,  60,  70,  80,  90,  200,  400,  600,  800, 
and  900. 

3.  Express  by  means  of  Eoman  numerals  the  numbers 
39,  49,  59,  69,  79,  89,  99,  96,  444,  1294,  and  1889. 

4.  Write  the  numbers  LVIII,  XXXIX,  XLIV,  XCIV, 
XCIX,  CXCIX,  and  MMDCCXCIX,  in  the  Arabic 
notation. 


Arts.  23,  24.]  ADDITION.  13 


CHAPTER   II. 

ADDITION—  SUBTRACTION  —  MULTIPLICATION  — 
DIVISION. 

Addition. 

23.  The  process  of  finding  a  single  number  which 
contains  as  many  units  as  there  are  in  two  or  more  given 
numbers  taken  together  is  called  Addition ;  and  this  single 
number  is  called  the  Sum. 

The  sum  of  the  numbers  of  the  units  in  two  or  more 
groups  would  therefore  be  found  by  forming  a  single 
group  containing  them  all,  and  then  counting  the  number 
of  the  units  in  this  single  group. 

24.  The  following  fundamental  truth  is  evident : 

TJie  number  of  the  things  in  any  group  will  always  be 
found  to  be  the  same  in  whatever  order  they  may  be  counted. 

From  this  it  follows  that  the  sum  of  the  numbers  of  the 
things  in  any  two  groups  will  be  found  by  first  counting 
all  the  things  in  the  first  group  and  then  proceeding  to 
the  second ;  that  is,  by  increasing  the  number  in  the  first 
group  by  as  many  units  as  there  are  in  the  second.  The 
same  sum  will  also  be  found  by  increasing  the  number  in 
the  second  group  by  as  many  units  as  there  are  in  the  first. 

Thus,  the  sum,  of  3  and  5  is  found  by  counting  five 
onwards  from  three,  namely  four,  five,  six,  seven,  eight; 
or  by  counting  three  onwards  from  five,  namely  six,  seven, 


14  ADDITION.  [Chap.  II. 

eight  In  the  first  case  we  are  said  to  add  5  to  3,  and  in 
the  second  case  we  are  said  to  add  3  to  5 ;  but  the  results 
must  be  the  same. 

25.  Addition  is  indicated  by  the  sign  -f,  which  is  read 
'  plus.* 

Thus,  5  +  4  is  read  five  plus  four,  and  denotes  that  5  is  to  be 
increased  by  4,  that  is,  that  4  is  to  be  added  to  5  ;  also,  5  +  4  +  3 
denotes  that  4  is  to  be  added  to  5,  and  then  3  added  to  the  result. 

26.  The  sign  =  ,  which  is  read  'equals'  or  'is  equal  to,' 
is  used  to  denote  the  equality  of  two  numbers. 

Thus,  5  +  4  =  9  denotes  that  the  sum  of  5  and  4  is  9. 

27.  When  children  first  begin  to  add  they  make  use 
of  their  fingers,  but  all  counting  on  the  fingers,  or  with 
any  other  real  objects,  should  be  discontinued  as  soon  as 
possible,  and  the  results  of  adding  numbers  not  greater 
than  nine  should  be  given  instantaneously. 

Tables  of  the  results  of  the  addition  of  any  two  numbers 
each  not  greater  than  10  might  at  first  be  made  by  the 
pupil,  arranged  in  lines ;  as  for  example,  8  and  1  are  9, 

8  and  2  are  10,  8  and  3  are  11,  etc. 

EXAMPLES  III. 
Oral  Exercises. 

These  examples  should  be  practised  until  great  rapid- 
ity is  attained. 

1.  Add  1  and  9,  3  and  8,  2  and  6,  4  and  7,  6  and  3, 
4  and  4. 

2.  Add  7  and  8,  7  and  6,  3  and  9,  5  and  4,  3  and  5, 

9  and  8. 

3.  Add  4  and  3,  9  and  9,  8  and  8,  6  and  9,  7  and  2, 
3  and  3. 


Arts.  25-28.]  EXAMPLES.  15 

4.  Add  5  and  9,  9  and  4,  6  and  8,  5  and  7,  2  and  9, 

8  and  5. 

5.  Add  7  and  7,  5  and  5,  6  and  6,  8  and  4,  6  and  4, 

9  and  7. 

6.  Add  8  to  15,  to  25,  to  35,  to  45,  to  65,  and  to  95. 

7.  Add  13  and  7,  23  and  7,  43  and  7,  63  and  7,  83 
and  7,  93  and  7. 

8.  Add  9  to  17,  to  27,  to  57,  to  67,  to  87,  and  to  97. 

9.  Begin  with  7  and  add  2  again  and  again  up  to  27. 

Do  not  say  7  and  2  are  9  and  2  are  11  and  2  are  13,  etc.,  but 
state  results;  thus,  7,  9,  11,  13,  etc. 

10.  Begin  with  2  and  add  3  again  and  again  up  to  35. 

11.  Begin  with  85  and  add  4  again  and  again  up  to 
101. 

12.  Begin  with  50  and  keep  on  adding  sevens  until 
the  sum  exceeds  100. 

13.  Begin  with  15  and  keep  on  adding  nines  until  the 
sum  exceeds  100. 

14.  Add  the  following  numbers  in  order,  first  begin- 
ning at  the  right  and  then  at  the  left : 

(1)  2,  7,  4,  0,  6,  9,  5,  2,  6,  5,  9,  3,  4,  8. 
State  results  only ;  thus,  2,  9,  13,  13,  19,  28,  etc. 

(2)  7,  9,  5,  4,  0,  8,  6,  7,  3,  5,  9,  8,  2,  6. 

(3)  3,  5,  6,  9,  0,  7,  8,  4,  3,  6,  2,  5,  7,  9. 

(4)  9,  6,  7,  4,  2,  8,  1,  3,  7,  5,  4,  6,  5,  8. 

28.  The  sum  of  any  two  numbers  may  be  found  by 
counting  onwards  from  the  first  as  many  units  as  there 
are  in  the  second,  but  this  method  would  obviously  be 
very  troublesome  except  when  the  second  number  is  very 
small. 


16  ADDITION.  [Chap.  II. 

Now  numbers  are  divided,  as  we  have  already  learned, 
into  groups  of  units,  tens,  hundreds,  tenths,  hundredths, 
etc. ;  and  when  numbers  are  to  be  added,  the  parts  into 
which  they  are  divided  may  be  added  in  any  order  we 
please,  provided  they  are  all  counted;  hence  we  may  first 
add  the  units  of  one  order,  then  the  units  of  another  order, 
and  so  on. 

29.  In  order  to  add  numbers,  they  should  first  be 
arranged  so  that  their  decimal  points  are  in  a  vertical 
column.  This  will  ensure  that  all  the  tenths  shall  be  in 
the  same  vertical  column,  and  so  for  the  hundredths, 
etc. ;  and  so  also  for  the  units,  tens,  hundreds,  etc.  This 
arrangement  is  for  convenience  only. 

The  following  examples  will  show  how  this  principle 
enables  us  readily  to  find  the  sum  of  any  given  numbers. 

Ex.  1.   Add  235.7  and  524.2. 

Since  we  wish  to  add  the  tenths  by  themselves,  the  units  by 
themselves,  etc. ,  we  write  the  numbers  so  that  the  decimal  points 
are  in  a  vertical  column  ;  thus, 

235.7 
524.2 

Now  2  tenths  and  7  tenths  make  9  tenths,  4  units  and  5  units 
make  9  units,  2  tens  and  3  tens  make  5  tens,  and  5  hundreds  and  2 
hundreds  make  7  hundreds.  The  required  sum  is  generally  placed 
just  under  the  numbers  to  be  added  and  separated  from  them  by  a 

horizontal  line  ;  thus, 

235.7 
524.2 
759.9 

Ex.  2.  Add  548.6,  789,  and  197.8. 

Write  the  numbers  as  in  Ex.  1  ;  thus, 

548.6 

789. 

197.8 


1535.4 


Arts.  29, 30.]  EXAMPLES.  17 

Now  8  tenths  and  6  tenths  make  14  tenths,  that  is,  1  unit  and  4 
tenths.     The  4  tenths  can  be  put  in  the  column  for  tenths,  but  the 

1  unit  must  be  counted  with  the  other  units.  We  then  have  1  unit, 
7  units,  9  units,  and  8  units,  which  make  25  units  ;  that  is,  2  tens 
and  5  units.  The  5  is  put  in  the  column  for  units,  but  the  2  tens 
are  '  carried '  (as  it  is  called)  and  added  with  the  other  tens.  So  we 
proceed  until  all  the  columns  are  added. 

Note.  Since  ten  units  of  any  order  make  one  unit  of  the  next 
higher  order,  the  figures  in  any  column  may  be  added  without 
specifying  the  kind  of  units  they  represent ;  that  is,  without  call- 
ing them  tens,  or  hundreds,  or  thousands,  etc.,  as  the  case  may  be. 

Also,  we  should  never  use  as  many  words  as  in  the  above  explana- 
tions, but  should  say  (see  ex.  2)  only  8,  14  ;  1  (carried),  8,  17,  25 ; 

2  (carried),  11,  19,  23  ;  2  (carried),  3,  10,  15.  Of  course  the  4,  5,  3, 
and  15  are  the  figures  to  be  written.  In  all  cases  the  sums  of  num- 
bers should  be  more  prominent  than  the  numbers  themselves. 

30.  To  detect  mistakes  in  addition  it  is  well  to  add 
each  line  of  figures  twice,  once  from  bottom  to  top  and 
once  from  top  to  bottom.  An  error  is  much  more  likely 
to  be  detected  in  this  way  than  by  simply  repeating  the 
addition  in  the  same  order,  for  the  same  mistake  is  very 
likely  to  be  made  again. 

Pupils  should  not  be  allowed  to  add  more  than  one 
column  at  a  time. 

EXAMPLES  IV. 
Written    Exercises. 

1.  Add  3104,  297,  5649,  and  989. 
Find  the  sum  of 

2.  21.63,  5.24,  170.63,  27.59,  17. 

3.  301.7,  30.17,  3.017,  .3017,  .03017. 

4.  319,  562,  1230,  857,  4908,  and  9087. 

5.  235,  796,  804,  987,  359,  and  856. 

6.  170.2,  3.605,  17.35,  15.609,  .0086. 


18 


ADDITION. 


[Chap.  II. 


7.  .0037,  21.85,  169.4,  17.9375,  .90087. 

8.  4.1372,  41.372,  4137.2,  .41372,  41372. 

Add  the  numbers  in  each  column  and  in  each  row  of 
the  squares.     Do  not  change  the  positions  of  the  numbers. 

9. 


1 

15 

14 

4 

10. 

9 

16 

23 

12 

5 

12 

6 

7 

9 

13 

2 

10 

19 

21 

8 

10 

11 

5 

20 

24 

11 

3 

7 

13 

3 

2 

16 

1 

8 

17 

25 

14 

22 

15 

4 

6 

18 

11. 


1 

101 

80 

59 

38 

17 

117 

96 

75 

54 

33 

21 

121 

89 

68 

47 

26 

5 

105 

84 

63 

42 

30 

9 

109 

88 

56 

35 

14 

114 

93 

72 

51 

39 

18 

118 

97 

76 

55 

23 

2 

102 

81 

60 

48 

27 

6 

106 

85 

64 

43 

22 

111 

90 

69 

57 

36 

15 

115 

94 

73 

52 

31 

10 

110 

78 

77 

45 

24 

3 

103 

82 

61 

40 

19 

119 

98 

86 

65 

44 

12 

112 

91 

70 

49 

28 

7 

107 

95 
104 

74 

83 

53 
62~ 

32 
41 

11 

20 

100 
120 

79 
99 

58 
67~ 

37 
^6~ 

16 
25 

116 
4 

113 

92 

71 

50 

29 

8 

108 

87 

66 

34 

13 

Perform  the  additions  indicated  below : 


12. 


3157 

13.      589.761 

14.   412.64506 

294 

35.71 

39.17412 

16903 

840.693 

246.82441 

8057 

392.75 

49.1733 

62934 

1569.4242 

387.198207 

998 

359.177 

129.38946 

JT.  31.] 

EXAMPLES. 

1! 

15.  50971 

16. 

314569 

17.  842713 

8265 

73985 

9185 

13926 

387648 

38977 

78912 

930807 

796359 

34056 

186794 

246824 

19389 

389548 

135791 

8747 

153875 

924678 

In  the  next  three  examples  do  not  change  the  positions 
of  the  numbers. 

18.  Find  30.1  +  297  +  35.16  + 1079  +  8.017  +  10.053. 

19.  Find  93084  +  15614  +  3801.76  +  536174  +  123456 
+  40.404. 

20.  Find  218904  +  37.215  +  .199  +  582163  +  397157 
+  81.429  +  7.9163. 

21 .  Add  six  hundred  ninety-five,  one  thousand  seventy- 
four,  eleven  thousand  four  hundred  eighty-nine,  and  fifty- 
four  thousand  three  hundred  seventy. 

22.  Add  three  million  four  hundred  seventeen  thou- 
sand thirty-five,  nine  hundred  forty-six  thousand  seven 
hundred,  fifteen  million  fifteen  thousand  fifteen,  and  sixty 
million  sixteen  hundred  twenty-four. 

23.  Add  six  million  five  hundred  nine  thousand  seven 
hundred  six  and  twelve  thousand  four  hundred  thirty- 
two  hundred-thousandths,  three  hundred  ninety  thousand 
and  four  hundred  twelve  thousandths,  eighteen  million 
forty  and  six  ten-thousandths. 

31.  Thus  far  we  have  studied  numbers  without  refer- 
ence to  objects. 

When  numbers  are  used  without  reference  to  any 
particular  units,  they  are  called  Abstract  Numbers. 

Two  and  five  are  abstract  numbers. 


20  ADDITION.  [Chap.  II. 

When  numbers  are  associated  with  particular  units, 
they  are  called  Concrete  Numbers. 

Two  feet  and  five  tons  are  concrete  numbers. 

32.  Concrete  numbers  can  be  added  only  when  the  unit 
is  the  same.  For  example,  3  horses  and  4  cows  do  not 
make  7  horses  nor  7  cows  ;  they  do,  however,  make  7 
animals;  because  regarding  them  as  animals  the  unit  is 
the  same.  Also  the  sum  of  3  feet  and  4  inches  is  not  7 
feet  nor  7  inches. 

EXAMPLES   V. 
Written  Exercises. 

1 .  In  1890  the  population  of  each  of  the  New  England 
States  was  as  follows  :  Maine,  661000 ;  New  Hampshire, 
377000;  Vermont,  332000;  Massachusetts,  2239000; 
Ehode  Island,  346000  ;  Connecticut,  746000.  What  was 
the  total  ? 

2.  In  a  town,  noted  for  the  number  of  its  schools, 
there  were  225  boys  in  a  military  school,  175  girls  in  a 
school  for  girls,  126  young  men  in  a  theological  school, 
163  boys  in  a  training  school,  23  children  in  a  kinder- 
garten, and  1500  pupils  in  the  public  schools.  How  many 
pupils  in  all  ? 

3.  A  man  paid  527.37  dollars  for  14  cows,  1463.80 
dollars  for  twelve  horses,  and  918.36  dollars  for  153  pigs. 
How  many  animals  were  there,  and  how  much  was  paid 
for  them  all  ? 

4.  The  population  of  each  of  the  six  northern  counties 
of  England  is  as  follows  :  Cumberland,  250647  ;  Durham, 
867258  ;  Lancashire,  3454441 ;  Northumberland,  434086  ; 
Westmoreland,  64191;  and  Yorkshire,  2886564.  What 
is  the  total  population  ? 


Arts.  32-37.]  SUBTRACTION.  21 


Subtraction. 

33.  The  process  of  finding  how  many  units  are  left 
when  a  number  is  taken  aivay  from  a  larger  number  is 
called  Subtraction.  The  result  is  called  the  Remainder, 
or  the  Difference. 

Any  two  numbers  can  be  added ;  it  is,  however,  impossible  to 
subtract  one  number  from  a  smaller  number. 

34.  The  larger  of  the  two  numbers  is  called  the 
Minuend. 

The  smaller  of  the  two  numbers  is  called  the  Sub- 
trahend. 

Illustration.  8    Minuend. 

5     Subtrahend. 
3    Remainder. 

35.  It  is  clear  that  the  remainder  is  that  number  ivhich, 
when  added  to  the  subtrahend,  will  give  the  minuend. 

Thus,  to  subtract  5  from  12  is  to  find  the  number  which,  when 
added  to  5,  will  make  12. 

The  question  involved  in  subtraction  may  be  put  in  different 
ways.     Thus,  it  may  be  asked  : 

(1)  What  is  the  remainder  when  5  is  taken  from  12  ? 

(2)  What  must  be  added  to  5  to  make  12  ? 

(3)  By  how  many  is  12  greater  than  5  ? 

(4)  By  how  many  is  5  less  than  12  ? 

36.  Subtraction  is  indicated  by  the  sign  — ,  which  is 
read  'minus.' 

Thus,  9  —  4  is  read  nine  minus  four,  and  denotes  that  9  is  to  be 
diminished  by  4,  that  is,  that  4  is  to  be  subtracted  from  9 ;  also, 
5  —  4  +  3  denotes  that  4  is  to  be  taken  from  5,  and  then  3  added 
to  the  result. 

37.  The  knowledge  of  the  results  of  the  addition  of 
numbers  not  greater  than  ten  will  furnish  us  with  the 


22  SUBTRACTION.  [Chap.  II. 

results  of  the  subtraction  of  small  numbers.  Examples 
of  subtractions  of  this  kind  should  be  practised  until 
great  rapidity  is  attained. 

EXAMPLES    VI. 
Oral  Exercises. 

1.  How  many  are  left  when  we  take  7  from  14, 
5  from  10,  6  from  12,  8  from  12,  4  from  10,  and  7  from 
16,  respectively  ? 

2.  How  many  are  left  when  we  take  5  from  14, 
4  from  13,  8  from  14,  7  from  12,  9  from  11,  and  5  from 
13,  respectively  ? 

Find  the  difference  between  the  numbers  in  each  of  the 
following  pairs : 

3.  5  and  12,  7  and  16,  9  and  18,  3  and  11,  6  and  14, 

8  and  15. 

4.  3  and  8,  5  and  11,  6  and  13,  8  and  14,  7  and  15, 

9  and  16. 

5.  Begin  with  50  and  go  on  diminishing  by  fours  as 
many  times  as  possible. 

6.  Begin  with  53  and  go  on  diminishing  by  fives  as 
many  times  as  possible. 

7.  Begin  with  70  and  go  on  diminishing  by  sixes  as 
many  times  as  possible. 

8.  What  must  be  added  to  5  to  make  8,  to  make  13, 
to  make  10,  to  make  12  ? 

9.  What  must  be  added  to  7  to  make  9,  to  make  12, 
to  make  10,  to  make  15  ? 

10.    What  must  be  added  to  8  to  make  10,  to  make  12, 
to  make  14,  to  make  16  ? 


Art.  38.] 


SUBTRACTION. 


23 


11.    9  and 
4  and 


Fill  up  the  blanks  below. 

make  10,  3  and 
make  11,  6  and 

12.  7  and     make  15,  6  and 
4  and    make    9,  3  and 

13.  3  and     make    7,  9  and 
6  and    make  15,  7  and 


make  11,  2  and 
make    9,  4  and 

make  13,  9  and 
make    8,  8  and 

make  18,  8  and 
make    9,  3  and 


make    8 
make    8 

make  12 
make  17 

make  16 
make    6 


38.  The  consideration  of  the  following  examples  will 
show  how  the  difference  between  any  two  numbers  can 
be  found. 

Ex.  1.     Subtract  524.63  from  759.85. 

The  smaller  number  should  be  placed  just  under  the  greater,  so 
that  one  decimal  point  is  vertically  over  the  other.     (See  Art.  29.) 

759.85 
524.63 


Beginning  with  the  lowest  order,  we  find  the  remainder  when 

3  hundredths  are  taken  from  5  hundredths,  6  tenths  from  8  tenths, 

4  units  from  9  units,  2  tens  from  5  tens,  and  5  hundreds  from 
7  hundreds ;  thus, 

759.85    Minuend. 
524.63     Subtrahend. 
235.22     Remainder. 

Ex.  2.     Subtract  35.7  from  78.3. 

78.3    Minuend. 
35.7     Subtrahend. 
42.6     Remainder 

Now  7  tenths  are  more  than  3  tenths,  therefore  we  cannot  sub- 
tract :  if,  however,  we  take  1  unit  from  the  8  units  and  change 
that  unit  to  10  tenths,  we  shall  have  13  tenths  in  all.  Now  7  tenths 
from  13  tenths  leave  6  tenths,  5  units  from  7  units  leave  2  units, 
and  3  tens  from  7  tens  leave  4  tens.     Remainder  =  42.6. 


24  SUBTRACTION.  [Chap.  II. 

Mental  Work  Illustrated.    We  may  omit  names  of  orders.    (See 
note,  Art.  29.) 

Ex.  3. 


468.27 
186.49 
281.78 

9  from  17, 
4  from  11, 
6  from    7, 

8. 
7. 
1. 

8  from  16, 

8. 

1  from    3, 

2. 

20.07 
12.6 

7.47 

0  from  7, 
6  from  10, 
2  from    9, 

7. 
4. 

7. 

1  from    1, 

0. 

Ex.  4. 


In  this  example  1  ten  is  taken  from  2  tens  and  changed  to  10 
units  ;  one  of  these  units  is  changed  to  ten  tenths.  The  operation 
may  be  represented  thus  : 

20.07  =19.  10  7 

12.6  =12.6 


Remainder  =      7.  4     7 

39.  One  concrete  number  cannot  be  subtracted  from  another 
unless  both  are  expressed  in  terms  of  the  same  unit.  For  example, 
we  cannot  subtract  5  tons  from  7  miles ;  nor  can  we  subtract  3 
feet  from  60  inches,  unless  either  3  feet  is  expressed  in  inches  or 
60  inches  expressed  in  feet. 

40.  It  is  easily  seen  that  if  from  a  given  number  several  num- 
bers be  taken  in  succession  the  result  will  be  the  same  as  if  the 
sum  of  those  numbers  were  subtracted  from  the  given  number. 

Ex.   Subtract  the  sum  of  366,  648,  and  759  from  2314. 

~^*  9,  8,  and  6  make  23 ;  subtract  the  3  from  the  4  and  carry 

j?^j  the  2  ;   2,  5,  4,  and  6  make  17  ;   subtract  the  7  from  11 

759  and  carry  the  1 ;  1,  7,  6,  and  3  make  17,  which  is  to  be  sub- 

541  tracted  from  22. 

Mental  Work. 
9,  17,  23,  3  from    4  =  1. 

2,7,11,17,  7     "     11=4. 

1,  8,  14,  17,  17     "     22  =  5. 


Arts.  39-42.]  EXAMPLES.  25 

41.  When  several  operations  of  addition  and  subtraction  have 
to  be  performed  in  succession  the  result  is  the  same  in  whatever 
order  the  operations  are  performed. 

Hence,  to  find  28  -  15  +  26  -  17  -  14  +  12,  first  find  the  sum  of 
28,  26,  and  12,  the  numbers  to  be  added  ;  then  the  sum  of  15,  17, 
and  14,  the  numbers  to  be  subtracted  ;  and  finally  taking  the 
difference  of  these  two  sums  ;  thus, 

28  15 
26  17 
12  14 
66  -  46  =  20. 

42.  To  detect  mistakes  in  subtraction,  add  the  remain- 
der to  the  subtrahend,  and  the  sum  should  equal  the 
minuend ;  or  subtract  the  remainder  from  the  minuend, 
and  the  new  remainder  should  equal  the  subtrahend. 


EXAMPLES   VII. 
Written  Exercises. 

1.  Subtract   129.6   from   3145,    81.7   from   3002,  and 
123.4  from  432.1. 

2.  Subtract  15.97  from   79.15,  18235   from   1000000, 
and  135.79  from  24680.6. 

3.  Find  the  values  of  645-378,  307-149,  294- 
208,  2179  -  1984,  3206  -  1679,  and  120573  -  98765. 


Fi 

nd  the  difference  between 

4. 

3.726  and  5.949. 

8.   3.008  and  3.08. 

5. 

14.753  and  6.876. 

9.    .217  and  .271. 

6. 

1  and  .888. 

10.    20  and  .675. 

7. 

.00013  and  .00175. 

11.    .8017  and  .00693. 

12.    Find  the  values  of 

(1)  31  +  97  -  23  +  175  -  184. 

(2)  151  -  77  +  94  -  111. 


26  SUBTRACTION.  [Chap.  II. 

(3)  315  -  127  -  172  +  358  -  265. 

(4)  742  -  329  -  197  +  215. 

13.  Find  3.17  +  4.216  -  5.8004  +  2.0097  -  .99873. 

14.  Find  21.09  -  3.985  -  7.0095  +  .09372  -  4.38009  + 
2.60009. 

15.  Subtract  from  11.214  the  sum  of  2.301,  1.7293, 
2.0507,  and  3.62743. 

16.  Subtract  from  20  the  sum  of  3.416,  2.6008,  5.73124, 
and  1.5063. 

17.  Subtract  from  121097  the  sum  of  7916,  1214, 1397, 
and  34162. 

18.  Subtract  from  1000000  the  sum  of  421654,  127, 
31562,  1795,  and  123456. 

19.  Subtract  27  from  80,  and  then  27  from  the  re- 
mainder, and  so  on  as  many  times  as  possible ;  and  find 
the  final  remainder. 

20.  What  number  must  be  taken  from  81  to  leave  37 
as  remainder  ? 

21.  By  how  much  does  the  sum  of  3.5612  and  4.71305 
exceed  the  sum  of  1.70862  and  5.91927? 

22.  What  number  must  be  taken  from  one  hundred 
thousand  to  leave  five  thousand  four  hundred  eighty- 
seven  as  remainder  ? 

23.  The  difference  between  two  numbers  is  145,  and 
the  greater  is  597 ;  what  is  the  smaller  ? 

24.  The  sum  of  two  numbers  is  1000,  and  one  of  them 
is  594 ;  what  is  the  other  ? 

25.  On  a  man's  birthday  in  1891  he  was  63  years  old. 
In  what  year  was  he  born  ? 

26.  In  1891  a  man  of  65  was  on  his  birthday  just  37 
years  older  than  his  son.     In  what  year  was  the  son  born  ? 


Arts.  43,  44.]  MULTIPLICATION.  27 

27.  Add  the  sum  of  516  and  784  to  the  difference  be- 
tween 314  and  176. 

28.  Add  the  difference  between  1925  and  1789  to  the 
difference  between  3421  and  1679. 

29.  In  an  orchard  there  are  1572  fruit  trees;  of  these 
352  are  apple  trees,  275  are  pear  trees,  and  187  are  plum 
trees.     How  many  other  trees  are  there  ? 

30.  The  population  of  each  of  five  towns  is  as  fol- 
lows: ^4,3789;  5,7861;  0,2893;  D,  756;  #,847.  If 
B  and  D  were  united,  the  new  town  would  be  how  much 
larger  than  A,  C,  and  E  together  ? 

Multiplication. 

43.  A  short  process  of  adding  two  or  more  equal  num- 
bers is  called  Multiplication. 

Ex.  1.   5  +  5  +  5  +  5  =  20  ;  i.e.,  4  fives  =  20. 

Ex.  2.  3  +  3  +  3  +  3  +  3  =  15  ;  i.e.,  5  threes  =  15. 

If  we  say  (Ex.  1)  5,  10,  15,  20,  or  (Ex.  2)  3,  6,  9,  12, 
15,  we  are  adding  by  a  long  process. 

If  we  say  4  fives  =  20,  or  5  threes  =  15,  we  are  adding 
by  .a  short  process  called  multiplication. 

44.  The  number  which  is  to  be  thus  increased  is  called 
the  Multiplicand. 

The  number  which  indicates  how  many  equal  numbers 
are  to  be  added  is  called  the  Multiplier. 

The  result  of  multiplication  is  called  the  Product. 

The  multiplicand  and  multiplier  are  called  Factors  of 
the  product. 

Ex.  1.  Multiply  5  by  4.  Ex.  2.  Multiply  3  by  5. 


5  Multiplicand.         3 

\  Multiplier.  _5 

20  Product.  15 


Factors  of  20  {   \  ^r*         !  !*  Factors  of  15. 

I   4  Multiplier.  5 


28 


MULTIPLICATION. 


[Chap.  II. 


45.  The  multiplication  of  any  two  numbers  not  greater 
than  nine  is  easily  found  by  actual  addition.  It  will  be 
shown  that  every  case  of  multiplication  can  be  reduced 
to  a  series  of  cases  of  multiplications  of  numbers  not 
greater  than  ten;  it  is  therefore  essential  to  learn  by 
heart  all  the  products  of  such  numbers.  These  products 
are  given  in  the  following  table,  called  the  Multiplication 
Table. 


1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

3 

6 

9 

12 

15 

18 

21 

24 

27 

30 

33 

36 

4 

8 

12 

16 

20 

24 

28 

32 

36 

40 

44 

48 

5 

10 

15 

20 

25 

30 

35 

40 

45 

50 

55 

60 

6 

12 

18 

24 

30 

36 

42 

48 

54 

60 

6Q 

72 

7 

14 

21 

28 

35 

42 

49 

56 

63 

70 

77 

84 
96 

8 

16 

24 

32 

40 

48 

56 

64 

72 

80 

88 

9 

18 

27 

36 

45 

54 

63 

70 

72 
80 

81 
90 

90 
100 

99 
110 

108 
120 

10 

20 

30 

40 

50 

60 

11 

22 

33 

44 

55 

66 

77 

88 

99 

110 

121 

132 
144 

12 

24 

36 

48 

60 

27 

84 

96 

108 

120 

132 

Any  horizontal  line  in  the  table  gives  the  products  of 
the  number  which  begins  the  line  by  the  first  twelve 
numbers  in  order.  Thus  the  fourth  line  can  be  read  1 
four  is  4,  2  fours  are  8,  3  fours  are  12,  4  fours  are  16,  etc. 

It  is  usual  and  desirable,  though  not  absolutely  necessary,  to 
learn  the  Multiplication  Table  as  far  as  12  times  12.     This  table 


Arts.  45-47.]  THEOREMS.  29 

should  be  made  again  and  again  by  the  pupil  himself  by  actual 
addition. 

46.  Multiplication  is  indicated  by  the  sign  x ,  which 
is  read  'multiplied  by'  or  'times.' 

Thus,  5  x?  4  is  read  5  multiplied  by  4,  which  means  4  times  5 ; 
also,  5x4x3  denotes  that  5  is  to  be  multiplied  by  4,  and  this 
product  multiplied  by  3. 

When  one  number  is  multiplied  by  two  or  more  other 
numbers  in  succession,  the  result  is  called  the  Continued 
Product. 

47.  Before  considering  how  to  find  the  product  of  any 
two  numbers,  certain  general  truths,  which  hold  good  for 
all  numbers  whatever,  must  be  investigated.  For  this 
purpose  consider  the  following  arrangement  of  dots : 


The  total  number  of  the  dots  is  independent  of  the  way 
in  which  they  are  counted. 

Now  there  are  10  dots  in  each  row  and  5  rows ;  the 
whole  number  of  the  dots  can  therefore  be  counted  as  10 
repeated  5  times,  or  6  repeated  10  times ;  i.e.,  10  x  5 
=  5  x  10.  It  is  clear  that  this  result  would  hold  good 
however  many  rows  and  columns  there  might  be ;  thus 
we  are  led  to 

Theorem  I.  The  product  of  any  number  by  any  second 
number  is  the  same  as  the  product  of  the  second  by  the 
first. 


30  MULTIPLICATION.  [Chap.  II. 

Again,  if  we  consider  separately  the  two  parts  divided 
by  the  vertical  line,  we  see  that  the  whole  number  of 
dots  is  the  sum  of  6  repeated  5  times  and  4  repeated  5 
times,  the  6  and  4  together  making  10,  so  that  10  x  5  is 
the  same  as6x5-f4x5;  thus  we  are  led  to 

Theorem  II.  The  product  of  any  two  numbers  is  the  same 
as  the  sum  of  the  products  of  the  multiplier  and  any  two  or 
more  numbers  ivhich  together  make  up  the  multiplicand. 

Now  consider  the  following  arrangement : 

5  5  5  5  5  5 
5  5  5  5  5  5 
5    5    5    5     5    5 

Here  the  sum  of  all  the  numbers  consists  of  5  repeated 
6x3  times.  But  in  each  row  5  is  repeated  6  times  and 
the  rows  are  repeated  3  times ;  hence  the  sum  of  all  the 
numbers  is  5x6x3.  Again,  in  each  column  5  is  re- 
peated 3  times  and  the  columns  are  repeated  6  times ; 
hence  the  whole  sum  is  5  x  3  x  6.  Thus,  to  multiply  by 
6  and  3  in  succession,  in  any  order,  gives  the  same  result 
as  to  multiply  at  once  by  6  x  3,  that. is  by  18.  It  is  clear 
that  the  same  would  be  true  for  any  other  numbers  what- 
ever ;  thus  we  are  led  to 

Theorem  III.  To  multiply  by  two  or  more  numbers  in 
succession  gives  the  same  result  as  to  multiply  at  once  by 
their  product. 

48.  To  multiply  by  10,  100,  etc.  Numbers  are  sepa- 
rated into  groups  of  units,  tens,  hundreds,  etc. ;  tenths, 
hundredths,  etc. ;  and  a  number  is  multiplied  by  10  when 
each  of  its  parts  is  multiplied  by  10  [Theorem  II] ;  i.e., 
when  each  of  its  parts  is  raised  to  the  next  higher  order. 

For  example,  62.34  is  multiplied  by  10  when  its  4  hundredths 
are  made  4  tenths,  its  3  tenths  are  made  3  units,  its  2  units  are 


Arts.  48,  49.]      TO   MULTIPLY  BY   10,    ETC.  gl 

made  2  tens,  and  its  6  tens  are  made  6  hundreds ;  this  is  accom- 
plished by  moving  the  decimal  point  one  place  to  the  right ;  i.e., 
62.34  x  10  =  623.4. 

Also,  623.4  x  10  =  6234  ;  6234  x  10  =  62340. 

Multiplying  by  10  and  then  by  10  again  is  the  same 
as  multiplying  by  100,  and  it  will  be  noticed  that  in  mul- 
tiplying 62.34  by  100,  the  decimal  point  is  moved  two 
places  to  the  right. 

Hence,  to  multiply  by  10, 100, 1000,  etc.,  move  the  decimal 
point  as  many  places  to  the  right  as  there  are  naughts  in  the 
multiplier. 

Examples.  1.   164.2789  x  10      =  1642.789. 

2.  164.2789  x  1000  =  164278.9. 

3.  340  x  100    =  34000. 

49.  The  following  examples  will  show  how  to  find  the 
product  of  any  two  numbers. 

Ex.  1.     Multiply  52.34  by  7. 

In  multiplication,  the  multiplier  is  placed  under  the  multiplicand, 
so  that  the  right-hand  figures  shall  be  in  the  same  vertical  column. 
This  is  only  for  convenience. 

52.34     Multiplicand. 
7     Multiplier. 
366.38    Product. 

By  Theorem  II,  we  multiply  the  units  of  the  different  orders 
separately  by  7,  and  add  the  results.  Now,  4  hundredths  x  7  =  28 
hundredths,  or  2  tenths  and  8  hundredths  ;  the  8  is  put  in  hun- 
dredths' column,  and  the  2  must  be  counted  with  the  tenths,  or 
'carried.'  Next,  3  tenths  x  7  =  21  tenths,  which  with  the  2  tenths 
carried  =  23  tenths,  or  2  units  and  3  tenths  ;  write  the  3  in  tenths' 
column,  and  carry  the  2  to  units'  column.  Next,  2  units  x  7  =  14 
units,  which  with  the  2  units  carried  =  16  units,  or  1  ten  and  6 
units  ;  write  the  6  in  units'  column,  and  cany  the  one  to  tens' 
column.  Finally,  5  tens  x  7  —  35  tens,  which  with  the  1  ten  carried 
=  36  tens  ;  write  the  5  in  tens'  column,  and  the  3  in  hundreds' 
column.     The  product  is  366.38. 


32  MULTIPLICATION.  [Chap.  II. 

Ex.  2.     Multiply  .3  by  .2. 

.3    Multiplicand. 
.2_    Multiplier. 
.06    Product. 

Now,  3  tenths  x  2  =  6  tenths.  Since  2  is  10  times  2  tenths, 
the  product  obtained  by  multiplying  by  2  is  10  times  the  product 
obtained  by  multiplying  by  2  tenths.  If,  therefore,  we  separate 
the  6  tenths,  which  equals  60  hundredths,  into  10  equal  parts, 
one  of  those  parts  must  be  the  true  product.  We  find,  from  the 
multiplication  table,  that  6  hundredths  is  one  of  the  10  equal  parts 
of  60  hundredths.     Therefore  .06  is  the  true  product. 

Ex.  3.  .08  x  3  =    .24  ;  .08  x  .3  =  .024. 

.18  x  4  =    .72  ;  .18  x  .4  =  .072. 
.53  x  2  =  1.06  ;  .63  x  .2  =  .106. 

Observe  that  the  number  of  decimal  places  in  the  product  is 
equal  to  the  number  of  decimal  places  in  both  multiplicand  and 
multiplier. 

Ex.  4.  Multiply  5.64  x  302.6. 

5.64      Multiplicand. 
202.6      Multiplier. 
1st  partial  product  3.384  =  5.64  x   6 

2d  partial  product  11.28    =  5.64  x    2 

3d  partial  product  1692.        =  5.64  x    2  x  100 

Sum  of  partial  products  1706.664  =  Product. 

We  multiply  5.64  separately  by  .6,  by  2,  and  by  200,  writing  the 
partial  products  in  form  for  addition;  i.e.,  so  that  the  decimal 
points  are  in  column. 

It  is  not  necessary  to  specify,  as  we  have  done  above,  the  kinds  of 
units  which  are  being  multiplied  at  any  stage.     (See  note,  Art.  29.") 

Ex.  5.    Multiply  321  x  218. 

321      Multiplicand. 
218      Multiplier 
2568  =  321  x  8 
3210  =  321  x  10 
64200  =  331  x  200 
69978  =  Product. 
The  naughts  are  omitted  in  practice,  because  they  count  for 
nothing  in  addition. 


Arts.  50-52.]  INVOLUTION.  33 

Ex.  6.             31642  Ex.  7.       27.006           Ex.  8.      27.68 

100506  2001.908  36000 

189852  21  6048  16608 

158210  24  30  54  8304 

31642  27  00  6  996480.00 

3180210852  54012      - 

.     54063.527448 

Naughts  at  the  right  of  multiplicand  or  multiplier  are  omitted 
in  the  partial  products  and  annexed  to  the  significant  figures  of 
the  answer  before  writing  the  decimal  point. 

50.  In  all  cases  of  multiplication  the  multiplier  must 
be  an  abstract  number,  for  to  repeat  anything  5  shillings 
times  or  3  tons  times  is  absurd;  the  multiplicand,  how- 
ever, may  be  either  a  concrete  number  or  an  abstract 
number. 

Thus,  we  can  multiply  3  feet  by  5,  but  not  3  feet  by  5  feet,  nor 
5  by  3  feet. 

In  case  a  person  is  reckoning  the  cost  of  5  pounds  of  tea  at  60 
cents  a  pound,  he  is  not  multiplying  60  cents  by  5  pounds,  but  is 
multiplying  60  cents  by  5,  since  he  is  finding  the  sum  of  as  many 
60  cents  as  there  are  pounds. 

51.  To  test  the  answer,  multiply  the  multiplier  by  the 
multiplicand.  This  should  give  the  same  result  as  mul- 
tiplying the  multiplicand  by  the  multiplier  [Theorem  I, 
p.  26]. 

52.  The  continued  product  of  a  number  by  itself  is 
called  a  Power  of  that  number. 

Thus,  5  x  5  is  called  the  second  power  or  the  square  of 
5;  5  x  5  x  5  is  called  the  third  power  or  the  cube  of  5; 
5x5x5x5  is  called  the  fourth  power  of  5 ;  and  so  on. 

The  squares  of  the  first  nine  numbers  are  1,  4,  9, 16,  25, 
36,  49,  64,  81.  The  cubes  are  1,  8,  27,  64,  125,  216, 
343,  512,  729. 

A  small  figure  placed  above  and  to  the  right  of  a 
number  is  used  to  denote  a  power  of  that  number,  and 


34  MULTIPLICATION.  [Chap.  II. 

is  called  an  Exponent,  or  Index.  For  example,  54  denotes 
5x5x5x5.  Also,  102  =  10  x  10  =  100,  103  =  10  x 
10  x  10  =  1000,  and  106  =  1000000. 

It  should  be  noticed  that  the  second  power  of  10  contains  two 
naughts,  the  third  power  three  naughts,  and  so  on. 

When  a  number  is  raised  to  a  power,  the  process  is 
called  Involution. 

Involution,  then,  is  but  a  name  for  a  special  case  of 
multiplication. 

EXAMPLES    VIII. 
Oral  Exercises. 

The  pupils  should  read  the  answers  while  looking  at 
the  examples. 

Multiply  each  of  the  following  by  10 : 

1.  23,  2.34,  51.67,  21.08,  700,  314.5. 

2.  769,  7.123,  .562,  1000.007,  .0034. 

3.  2.006,  .00006,  150.27,  5000000. 

Perform  the  multiplications  indicated  below : 

4.  2.157  x  100.  10.  .07  x  1000. 

5.  1.2308  x  1000.  11.  3.501  x  1000. 

6.  1.27  x  100.  12.  .00039  x  10000. 

7.  6573  x  100.  13.  98.764  x  1000. 

8.  .0067  x  100.  14.  .00001  x  1000. 

9.  2.1345  x  1000.  15.  16.02  x  100000. 

16.  Find  the  squares  of  4,  7,  2,  .3,  5. 

17.  Find  92,  .92,  122,  1.22,  .122. 

18.  Find  52,  .52,  .62,  .72,  .082. 

19.  Find  the  cubes  of  2,  3,  .2. 

20.  Find  .23,  .023,  .033,  .0033. 


RT.  52.] 

EXAMPLES 

3, 

EXAMPLES  : 

EX. 

Multiply 

Written  Exercises. 

I.  37  by  3. 

7. 

1083  by 

11. 

13.  5.7  x  8. 

2.  65  by  5. 

8. 

3408  by 

12. 

14.  31.09  x  23. 

3.  253  by  9. 

9. 

597  by  : 

LI. 

15.  1.25  x  7. 

4.  197  by  8. 

10. 

1.6  x  4. 

16.  3.72  x  9. 

5.  384  by  7. 

11. 

12.56  x 

17. 

17.  92.74  x  37. 

6.  909  by  6. 

12. 

142857  by  7. 

18.  7.314  x  84. 

19.  12345679  by  { 

d. 

39. 

125  by  47. 

20.  25  by  20. 

40. 

384  by  65. 

21.  27  by  30. 

41. 

908  by  73. 

22.  36  by  70. 

42. 

18  by  12345679. 

23.  318  by  50. 

43, 

63  by  12345679. 

24.  527  by  60. 

44. 

12.34  x  2.4. 

25.  894  by  80. 

45. 

38.24  x  3.9. 

26.  125  by  700. 

46. 

.1729  x  .24. 

27.  389  by  600. 

47. 

.3462  x  .75. 

28.  239  by  900. 

48. 

3.4165  x  3.57. 

29.  21670  by  4000. 

49. 

.2675  x  3.85. 

30.  5790  by  8000. 

50. 

697  by  123. 

31.  6175  by  8000. 

51. 

587  by  358. 

32.  821400  by  5000. 

52. 

399  by  586. 

33.  25  by  25. 

53. 

2809  by  702. 

34.  27  by  39. 

54. 

1973  by  904. 

35.  79  by  97. 

55. 

3097  by  807. 

36.  38  by  56. 

56. 

.0827  x  .2413. 

37.  79  by  87. 

57. 

.0237  x  .5214. 

38.  98  by  39. 

g 

58. 

3156  by  2065. 

36 


MULTIPLICATION. 


[Chap.  II. 


59.  .13579  x  .0246. 

60.  7802  by  2005. 

61.  2.31575  x  4.0824. 

62.  325.1  by  35.79. 

63.  .021628  x  .002828. 

64.  13579  by  21695. 

65.  .01  x  .01  x  .01. 

66.  .5  x  .05  x  .005. 

67.  6  x  .6  x  .06. 

68.  2.5  x  .25  x  .025. 

69.  3109.72  by  90.706. 

70.  823156  by  753698. 

71.  826075  by  1509607. 

72.  8257.314  by  78167094. 

73.  2178  by  5506. 

74.  3008  by  2345. 

75.  687.4  by  .2468. 

76.  12837  by  56294. 


77.  305.009  by  72809. 

78.  123.456  by  65.4321. 

79.  5009826  by  7090.068. 

80.  21840376  by  9287915. 

81.  3.9017  x  .215. 

82.  .00167  x  .0589. 

83.  2.1046  x  4.0035. 

84.  .21089  x  .003904. 

85.  .1  x  .1  X  .1. 

86.  .31  x  .41  x  .51. 

87.  20  by  125. 

88.  50  by  350. 

89.  800  by  125. 

90.  20  by  315. 

91.  400  by  125. 

92.  600  by  8745. 

93.  8000  by  1250. 

94.  12000  by  28971. 


Find  the  continued  products  of 

95.  12,  18,  and  15.  98.   10,  11, 12, 13,  and  14. 

96.  17,  18,  and  19.  99.   2,  .6,  73,  and  5. 

97.  3,  4,  5,  6,  7,  8,  and  9.     100.    .7,  .3,  .006,  and  1000. 

101.    Find  3  x  7  x  9  x  11  X  13  X  37. 


Find  the  squares  of 

102.  170.    103.  220.    104 


360. 


Find 

106.  1252. 

107.  5.372. 


108.  46902. 

109.  647002. 


105.  430. 

110.  174952. 

111.  215.7292. 


Arts.  53, 54.]  NUMBERS   IN   PARTS.  37 

112.  803.  116.  4253.  120.  .63. 

113.  1603.  117.  16083.  121.  .763. 

114.  8003.  118.  35073.  122.  .0063. 

115.  16003.  119.  167303. 

123.  There  are  2240  pounds  in  one  long  ton.  How 
many  pounds  are  there  in  517  long  tons  ? 

124.  There  are  168  hours  in  one  week.  How  many 
hours  are  there  in  506  weeks  ? 

125.  There  are  24  sheets  in  a  quire  of  paper,  and  20 
quires  in  a  ream.  How  many  sheets  are  there  in  524 
reams  ? 

126.  There  are  86400  seconds  in  a  day.  How  many 
are  there  in  365  days  ? 

127.  In  an  orchard  there  are  57  rows  of  gooseberry 
bushes,  and  there  are  256  bushes  in  each  row.  How 
many  bushes  are  there  altogether  ? 

128.  A  book  has  312  pages,  on  each  page  there  are  32 
lines,  and  in  each  line  there  are  42  letters.  How  many 
letters  are  there  altogether  ? 

53.  It  is  often  convenient  to  express  a  number  in  parts, 
connected  by  the  sign  -f  ;  thus,  we  may  write  8  -f  4,  or 
11  + 1,  or  7  +  3  +  2,  instead  of  12. 

54.  A  number  expressed  in  parts  may  be  multiplied  just 
as  if  expressed  as  a  whole ;  thus, 

~l~  m  Here  we  have  6  units  plus  5  units  to  be  multiplied  by 
42  _|_  35    7  J  the  answer  has  the  same  value  as  11  x  7. 

Note.  A  parenthesis  may  be  used  to  indicate  that  the  several 
parts  compose  one  number. 


94-    6 


38  MULTIPLICATION.  [Chap.  II. 

Again, 
9x4+3x8  +  10         Here  we  have  (9  x  4)  units  plus  (3  x  8) 

4     units  plus  10  units,  all  to  be  multiplied  by  4  ; 

144     +  96      +  40    the  answer  has  the  same  value  as  70  x  4. 

3  +  2  Again,  to  multiply  5  by  5,  or  to  find  5'2,  we  may 
3  +  2  multiply  3  units  plus  2  units  first  by  2  and  then  by  3 
6  +  4  and  add  the  partial  products  ;  the  answer  has  the  same 
value  as  5  x  5.  It  is  evident  that  we  may  easily  square 
9  +  12  +  4  any  number  between  10  and  100  after  separating  it 
into  two  parts,  —  its  tens  and  its  units.  For  example,  322  =  the 
square  of  32  after  being  expressed  as  the  sum  of  its  tens  and  its 
units. 

30  +  2 

30  +  2 

60  +  4 

900+    60 

900  +  120  +  4 


EXAMPLES   X. 
Written  Exercises. 

1.  Multiply  (9  +  8)  by  12 ;  (6  + 11)  by  8. 

2.  Multiply' (4  x  3  +  6)  by  7 ;  (9  x  2  +•  8  x  3)  by  2. 

3.  Multiply  (6  +  4  +  2)  by  3;  (80  +  4  x  8)  by  4. 

4.  Find  13* ;  212;  472;  942;  692. 

55.  The  following  methods  are  practical  and  are  of 
great  value  in  saving  time.  Pupils  should  become  pro- 
ficient in  performing  examples  by  these  methods,  and 
should  use  them  constantly. 

I.    Multiplication  table  for  numbers  between  12  and  20. 

Multiply  the  units  of  multiplicand  and  multiplier,  and 
write  the  unit  figure  of  the  product ;  then  add  the  tens 
figure  (if  any)  of  the  product,  the  multiplier,  and  the 
units  of  the  multiplicand.  , 


Art.  55.]  SHORT   PROCESSES.  39 


Ex.  1. 

16  Mental   Work. 

6  x  3  =  18 ;    write  8,  then   1  +  13  +  6  =  20 ;  write  20. 


13 

208 


Ex.2. 

13  Mental  Work. 

-JL3.        3x3  =  9:  write  9,  then  13  +  3  =  16 ;  write  16. 
16.9 

II.  To  multiply  by  11. 

Write  the  unit  figure  of  the  multiplicand;  then  add 
units  and  tens,  tens  and  hundreds,  etc.,  separately,  writ- 
ing the  right-hand  figures  of  the  several  sums  and  carry- 
ing the  left-hand  figures ;  finally,  write  the  last  figure  of 
the  multiplicand  after  adding  what  was  carried. 

7 

Ex.1.      469047  7  +  4  =  11 

11  1  (carried)  +4=   5 

5159517  9  +  lZll 

1  +  6  +  4  =  11 
1  +  4=   5 

The  last  figures  in.  the  column  are  the  ones  to  be  written  in  the 
product. 

Ex.  2.     89006.037  Ex.  3.        49.8769 

n  .11 

979066.407  5.486459 

Mental  Work.  Mental  Work. 

7,  10,  4,  6,  6,  0,  9,  17,  9.  9,  15,  14,  16,  18,  14,  5. 

III.  To  square  a  number  of  two  figures  and  ending  in  5. 
Square  the  units  and  write  the  whole  product;  then 

square  the  tens,  add  the  tens  to  this  square,  and  write  the 
sum. 

Ex.  1.  35  Ex.  2.        6.52  =  42.25. 

35  For  52  =  25 

1225  and      62+6=42. 

Here  5  x  5  =  25 
and  32  +  3  =  12, 


40  MULTIPLICATION.  [Chap.  II. 

EXAMPLES  XI. 

Multiply 

1.  14  by  18.  6.  19  by  19. 

2.  19  by  13.  7.  1.8  by  18. 

3.  1.6  by  1.3.  8.  .16  by  .16. 

4.  .18  by  1.5.  9.  6  by  3  by  14. 

5.  1.3  by  .13.  10.  2  by  8  by  19. 
Multiply  the  following  by  11 : 

11.  26751.  15.  64.9786.  19.  .463. 

12.  498.67.  16.  800960.  20.  590001. 

13.  94600.  17.  493.006.  21.  9000095. 

14.  888.  18.  2.76398.  22.  399678. 
Find  the  following : 

23.  552;  5.52;     .552.  26.    652;  7.52;    852. 

24.  152;  .152;  1502.  27.    252;  .252;    2502. 

25.  452;  4.52;  .0452.  28.    352;  3502;  35002. 

This  method  may  be  used  in  finding  the  squares  of  105, 
115,  and  125. 

After  a  little  practice,  much  of  the  above  work  may  be  done 
without  writing  anything  but  the  results. 

Division. 

56.  A  short  process  of  finding  out  how  many  equal 
numbers  may  be  together  subtracted  from  another  number 
is  called  Division. 

For  example,  to  divide  12  by  4  is  to  find  out  how  many  fours 
may  be  together  subtracted  from  12,  — to  find  out  how  many  fours 
there  are  in  12.  The  simplest  method  of  finding  the  required 
number  of  fours  is  to  subtract  4  from  12,  and  then  4  from  the 
remainder,  and  so  on,  as  many  times  as  possible.  It  will  be  found 
that  there  is  no  remainder  after  subtracting  3  fours.  Hence  there 
are  3  fours  in  12. 


Arts.  56-59.]  DIVISION.  41 

57.  The  number  which  is  to  be  thus  diminished  is 
called  the  Dividend. 

One  of  the  equal  numbers  to  be  subtracted  is  called 
the  Divisor. 

The  result  of  division  is  called  the  Quotient. 

Ex.  1.     Divide  24  by  8.  Ex.  2.     Divide  30  by  10. 

24  is  the  dividend.  30  is  the  dividend. 

8  is  the  divisor.  10  is  the  divisor. 

3  is  the  quotient.  3  is  the  quotient. 

58.  Since  the  dividend  contains  the  divisor  as  many 
times  as  there  are  units  in  the  quotient,  the  dividend  is 
equal  to  the  product  of  the  divisor  and  the  quotient. 

We  may  say  then  that  division  is  the  process  by  which 
one  factor  may  be  found  when  the  product  and  the  other 
factor  (or  factors)  are  given. 

Thus,  division  is  the  inverse  or  undoing  of  multiplica- 
tion, just  as  subtraction  is  the  undoing  of  addition. 

59.  Division  can  be  looked  upon  from  two  different 
points  of  view,  the  distinction  between  which  is  best 
seen  by  taking  as  an  example  the  division  of  a  concrete 
number. 

We  have  5  feet  x  7  =  35  feet ;  and  in  connection  with 
the  undoing  of  this  multiplication,  there  are  the  two 
distinct  questions: 

(1)  How  many  times  is  5  feet  contained  in  35  feet  ? 
The  answer  to  which  is  7  times. 

(2)  If  35  feet  be  divided  into  7  equal  parts,  what  will 
will  each  part  be  ?  Or,  what  length  is  contained  7  times 
in  35  feet  ? 

The  answer  to  which  is  5  feet. 

Thus,  in  division,  either  the  divisor  is  an  abstract 
number  and  the  quotient  a  quantity  of  the  same  kind  as 


42  DIVISION.  [Chap.  II. 

the  dividend,  or  else  the  divisor  is  a  quantity  of  the  same 
nature  as  the  dividend,  and  the  quotient  is  an  abstract 
number. 

60.  Division  is  indicated  by  the  sign  ^-,  which  is  read, 
' divided  by,'  or,  'by.' 

Thus,  24  h-  4  is  read  24  divided  by  4,  and  denotes  that  24  is  to  be 
divided  by  4  ;  also,  24  -+■  4  -4-  3  denotes  that  24  is  to  be  divided  by 
4  and  the  result  divided  by  3,  and  24  +  4  x  3  denotes  that  24  is  to 
be  divided  by  4  and  the  result  multiplied  by  3. 

61.  Inexact  Division.  If  we  try  to  divide  14  by  4,  we 
find  that  after  subtracting  3  fours  there  are  2  units  left. 

The  number  left  over  is  called  the  Remainder. 
One  number  is  said  to  be  exactly  divisible  by  another 
when  it  is  divisible  without  remainder. 

62.  It  follows  from  the  definition  of  division  that  the 
product  of  the  divisor  and  the  quotient  plus  the  remainder 
is  equal  to  the  dividend;  that  is, 

Divisor  x  Quotient  +  Remainder  =  Dividend. 
Hence,  if  any  three  of  these  four  numbers  be  given,  the 
remaining  one  can  be  found. 

Ex.  1.     The  divisor  is  5,  the  quotient  is  20,  and  the  remainder 

is  2.     What  is  the  dividend? 

The  dividend  must  exceed  the  product  of  the  divisor  and  quotient 

by  2.     Hence, 

Dividend  =  5  x  20  +  2  =  102. 

* 

Ex.  2.  The  dividend  is  59,  the  quotient  7,  and  the  remainder  3. 
What  is  the  divisor? 

The  dividend  must  exceed  the  product  of  the  quotient  and  divisor 
by  3.  Hence,  the  product  of  the  quotient  and  divisor  is  59  -  3  =  56, 
and  the  divisor  =  56  -*-  7  =  8. 

63.  Division  could  always  be  performed  by  successive 
subtractions  of  the  divisor,  as  in  Art.  56 ;  but,  except  in 


Arts.  60-64.]  DIVISION   BY  10,  ETC.  43 

the  case  of  very  small  numbers,  the  process  would  be 
extremely  tedious,  and  the  necessity  for  these  successive 
subtractions  is  obviated  by  a  knowledge  of  the  results  of 
multiplication. 

For  example,  to  divide  75  by  9. 

Since  we  know  that  8  nines  are  72,  and  that  9  nines  are  81,  we 
see  that  75  -f-  9  gives  8  for  quotient  and  3  for  remainder. 

EXAMPLES  XII. 
Oral  Exercises. 

Give  the  quotient  in  each  of  the  following  cases,  and 
the  remainder  whenever  the  division  is  not  exact : 

1.  12-4.  11.  64-8.  21.  80-9. 

2.  18-9.  12.  45-9.  22.  55-9. 

3.  35-7.  13.  15-4.  23.  53-7. 

4.  56-1-8.  14.  17^5.  24.  48-5. 

5.  60-10.  15.  18-7.  25.  92-9. 

6.  49-7.  16.  17-3.  26.  87-8. 

7.  81-9.  17.  37 -h  9.  27.  80-7. 

8.  72-8.  18.  43 -=-5.  28.  63-5. 

9.  56-^-7.  19.  68-7.  29.  70-6. 
10.  36-^-6.  20.  70-8.  30.  100-9. 

64.  Division  by  10,  100,  etc.  To  divide  any  number 
by  10,  it  is  necessary  only  to  move  the  decimal  point  one 
place  to  the  left.  For  this  divides  each  of  the  parts  of 
the  number  by  10. 

For  example,  623.4  (see  ex.,  Art.  48)  is  divided  by  10  when  its 
6  hundreds  are  made  6  tens,  its  2  tens  are  made  2  units,  its  3  units 
are  made  3  tenths,  and  its  4  tenths  are  made  4  hundredths ;  i.e., 
623.4  -  10  =  62.34. 

Also,  62.34  ■*■  10  =  6.234  ;  6.234  -  10  =  .6234. 


44  DIVISION.  [Chap.  II. 

Dividing  by  10  and  by  10  again  is  the  same  as  divid- 
ing by  100,  and  it  will  be  noticed  that  in  dividing  623.4 
by  100  the  decimal  point  is  moved  two  places  to  the  left. 

Hence,  to  divide  by  10,  100,  1000,  etc.,  move  the  decimal 
point  as  many  places  to  the  left  as  there  are  naughts  in  the 
divisor. 

Examples.  1.   268706  -=-10      =  26870.6. 

2.  46000  *  100    =      460. 

3.  26783  -  1000  =       26.783. 

65.  Short  Division. — When  the  divisor  is  not  greater 
than  12,  the  process  of  division  can  be  written  in  a 
very  compact  form.  The  method  will  be  seen  from  the 
following  example : 

Ex.    Divide  43251  by  8. 

The  operation  is  set  down  in  the  following  form  : 

8  )  43251 

5406,  remainder  3. 

Explanation.  First,  43  -=-  8  gives  quotient  5  and  remainder  3  ; 
we  pat  5  under  the  3  of  the  dividend,  as  the  5  represents  units  of 
the  same  order  as  the  3  (namely,  thousands',  in  the  present  case). 
Then,  the  remainder  3  is  equal  to  30  units  of  the  next  lower  order, 
and  taking  into  account  the  next  figure  of  the  dividend,  namely  2, 
we  have  32  which  when  divided  by  8  gives  quotient  4  and  0 
remainder  ;  we  put  down  4  next  to  5,  and  have  nothing  to  '  carry.' 
Then,  5  -=-  8  gives  quotient  0  and  remainder  5 ;  we  put  down  0  next 
to  4  and  'carry'  5.  The  5  carried  and  1,  the  next  figure  of  the 
dividend,  make  51  which  when  divided  by  8  gives  quotient  6  and 
remainder  3.    Thus,  the  complete  quotient  is  5406  with  remainder  3. 

EXAMPLES  XIII. 
Written  Exercises. 

Divide 

1.  92  by  4.      3.  75  by  5.      5.  7.85  by  5. 

2.  87  by  3.      4.  234  by  6.     6.  91.8  by  9. 


Arts.  65,  66.]  LONG   DIVISION.  45 

7.  72.15  by  5.  11.  7568  by  11.  15.  823507  by  8. 

8.  6.402  by  6.  12.  35.628  by  12.  16.  2104316  by  6. 

9.  .3564  by  9.  13.  72156  by  9.  17.  123456  by  7. 
10.  6822  by  12.  14.  346089  by  7.  18.  987654  by  9. 

19.   563753696  by  11.         20.    1374819756  by  12. 

Divide  without  uniting  the  terms  of  the  dividend 

21.  (8  +  14  +  6)  by  2. 

22.  (6  x  2  +  15x5)  by  3. 

23.  (14  x  3  +  21x5)  by  7. 

24.  (18  x  4  +  33  x  12)  by  3,  and  the  result  by  2. 

66.  Long  Division.  —  When  the  divisor  is  greater  than 
12  the  process  of  division  is  written  in  a  long  form  so 
that  the  mind  will  not  become  confused. 

Ex.  1. 


Divide  1026  by  18. 
18)1026(57 

The  full  operation  may  be 
thus  expressed : 

90 

126 

126 

18)1026(50  +  7 
900 
126 
126 

First,  beginning  at  the  left,  we  use  the  smallest  part  of  the  number 
that  can  be  divided  by  18.  Now,  neither  1  nor  10  can  be  divided 
by  18,  but  102  can  be.  102  -t- 18  =  6,  with  a  remainder  of  12.  The 
quotient  5  is  of  the  same  order  as  the  last  figure  of  the  dividend 
used  in  the  first  division  (just  as  in  short  division).  The  remainder 
12  we  reduce  to  units  of  the  next  lower  order  and  add  the  6  of  that 
order,  and  we  have  126  to  be  divided  by  18.     Now  126  -*- 18  =  7. 

Ex.  2.   Divide  102739  by  29. 


29)102739(3542 
87 

157  The  last  figure  of  the  dividend  used  in 

145  the  first  division  is  2,  and  in  thousands' 

J^3  place.     Therefore  the  first  quotient  figure 

obtained  is  thousands'. 


79 

58 

21  remainder. 


46  DIVISION.  [Chap.  II. 

Ex.  3.     Divide  44393  by  145. 

In  this  example,  the  first  remainder  (8 
i        l  hundreds)  reduced  to  tens  and  the  9  tens 

aqz.  added  makes  89  tens,  which  does  not  con- 

g^  tain  145.     Therefore  there  are  no  tens  in 

870  the  answer  and   we  write  a  naught,  and 

~23  remainder,     proceed  by  reducing  the  89  tens  to  units, 
adding  3  units. 

The  same  reasoning  applies  for  a  deci- 

'     '        1 — —         mal  dividend  as  for  an  integral  dividend. 

The  first  figure  obtained  in  the  quotient  is 

of  the  same  order  as   the   last  figure  used  in  the  first  division. 

This  fact  determines  the  position  of  the  decimal  point. 

Ex.  5.  Ex.  6.  Divide  .019  by  125. 

9). 288 


~032  125).019000(.000152 

125 
650 
625 
250 
250 

Ex.  7. 
48)5.220(.108 

— -  The  remainder  is  the  same  in  name  as  the  last 

381  figure  of  the  dividend.     In  this  case  it  is  .036. 

"36 

Ex.  8. 

15)474000(31600  In  this  example,  it  is  unnecessary  to  extend 

45  the   written   work  beyond  dividing    90   hun- 

^^  dreds  by  15.     Since,  however,  every  order  of 

the  dividend  must  have  a  corresponding  figure 

in  the  quotient,  we  write  naughts  in  tens'  and 

units'  places. 


90 
90 


67.    To  divide,  when  the  divisor  is  partly  or  wholly  a 
decimal. 


Art.  67.]  DIVISION   BY  A   DECIMAL.  47 

Here  we  make  use  of  the  following  principle : 

Multiplying  both  dividend  and  divisor  by  the  same  num- 
ber does  not  change  the  quotient. 

Thus,  24  -=-  4  =  6,  and  if  both  24  and  4  be  multiplied  by  2,  we 
shall  have  48  +  8  =  6  ;  also,  3.6  +  .6  =  6,  and  if  both  3.6  and  .6 
be  multiplied  by  10,  we  shall  have  36  -s-  6  =  6. 

Hence,  to  divide  any  number  by  a  decimal,  we  first 
multiply  both  dividend  and  divisor  by  that  power  of  10 
which  will  make  the  divisor  a  whole  number,  and  then  pro- 
ceed as  in  the  case  of  division  by  a  whole  number.  We 
perform  these  multiplications  by  moving  the  decimal 
points. 

Ex.  1.  Divide  11.68  by  1.6 

Move  the  decimal  points  one  place  to  the  right.     Then 

1.6J11.68(7.3 
112 

48 
48 

Do  not  forget  that  the  first  significant  figure  of  the  quo- 
tient is  of  the  same  order  as  the  last  of  those  figures  of  the 
dividend  which  are  used  in  the  first  division.  This  will 
indicate  the  position  of  the  decimal  point  in  the  quotient. 

Ex.  2.    Divide  .21  by  .0125  Ex.  3.    Divide  .0697585  by  1.33. 

.0125  ).2lJ^B.8  1.33J.06  97585C.05245 

125  ' 


850 
750 
100  0 
100  0 


6J35 
325 
266 
598 
532 
665 
665 


Note.    Always  let  the  old  decimal  point  remain,  and  indicate 
the  new  one  by  a  mark  similar  to  those  in  Ex.  1. 


48  DIVISION.  [Chap.  II. 

68.  Note.  It  should  be  noticed  that,  although  the  quotient  is 
unchanged  by  multiplying  both  dividend  and  divisor  by  the  same 
number,  the  remainder,  if  any,  is  not  unchanged,  but  is  equal  to 
the  original  remainder  multiplied  by  the  number  by  which  the 
original  divisor  and  dividend  were  multiplied. 

For  example,  26  -=-  6  =  4,  with  a  remainder  of  2  ;  and  8  times  26 
divided  by  8  times  6  equals  4,  with  a  remainder  of  8  times  2. 
Therefore,  we  must  divide  the  remainder  by  the  multiplier,  if  we 
wish  the  remainder  obtained  by  using  the  original  numbers,  as  the 
remainder  is  the  part  of  dividend  not  used. 

Ex.  1.  Divide  17.8  by  1.4. 


1.4J17.8v(12 

14 The  remainder  would  have  been  1 

3  8  unit  if  we  had  not  multiplied  by  10. 

28 

Remainder  1  0  units. 

Ex.  2.  How  many  pieces  each  1.02  inches  long  can  be  cut  from 
a  rod  ichose  length  is  18  inches  f 

We  can  find  the  quotient  by  dividing  1800  by  102.     Thus 

102)1800(17 
102 
780 
714 
66 

Hence  there  are  17  pieces ;  and  since  the  original  divisor  and 
dividend  were  multiplied  by  100,  the  remainder  left  over  is 

(6Q  ■*■  100)  inches  =  .66  inches. 

69.  Division  by  Factors.  — We  have  seen  that  to  multi- 
ply by  two  or  more  numbers  in  succession  gives  the  same 
result  as  to  multiply  at  once  by  their  product.  It  there- 
fore follows,  conversely,  that  to  divide  by  two  or  more 
numbers  in  succession  gives  the  same  result  as  to  divide 
at  once  by  the  product  of  the  numbers. 


Arts.  68-70.]  BY  FACTORS.  49 

Ex.  1.    Divide  11445  by  35, 

Since  35  =  7  x  5,  we  may  divide  by  7  and  5  in  succession. 
7)11445 
5)1635 
327 
Ex.  2.    To  divide  315637  by  20. 

20)31563J 

15781,    remainder  17. 

Dividing  both  dividend  and  divisor  by  10,  as  indicated,  we  have 
31563  to  be  divided  by  2.  The  quotient  is  15731  and  the  remainder 
1,  which  must  be  multiplied  by  10  and  the  figure  cut  off  by  the 
decimal  point  annexed,  making  17  as  the  true  remainder. 

70.  When  one  number  is  divided  by  several  others  in 
succession,  the  method  of  finding  the  remainder  will  be 
seen  from  the  following  example : 

Ex.  1.   Divide  11467  by  35. 
7)11467 

5)  1638  sevens  and  1  unit  over. 

327  thirty-fives  and  3  sevens  over. 
The  whole  remainder  is  therefore  3  sevens  and  1  unit,  that  is,  22. 

From  the  above  it  will  be  seen  that  the  whole  remain- 
der is  found  by  multiplying  the  remainder  after  the  second 
division  by  the  first  divisor  and  then  adding  the  remainder 
after  the  first  division. 

Ex.  2.   Divide  251633  by  8  x  6  X  7. 
3)251633 
5)83877  groups  of  3  each  and  2  units  over. 
7)16775  groups  of  3  x  5  each  and  2  groups  of  3  each  over. 
2396  groups  of  3  x  5  x  7  each  and  3  groups  of  3  x  5  each  over. 
The  whole  remainder  is  therefore  3  groups  of  3  x  5  each  +  2 
groups  of  3  each  +2  =  3x3x5  +  2x3  +  2  =  45 +  6  +  2  =  53. 

Thus,  if  there  are  more  than  two  successive  divisions  the  whole 
remainder  is  found  by  multiplying  each  remainder  by  all  the  di- 
visors preceding  that  from  which  the  remainder  arises,  and  then 
adding  these  results  to  the  first  remainder. 


50  DIVISION.  [Chap.  II. 

71.  The  work  of  finding  some  products  may  be  short- 
ened by  making  use  of  multiplication  and  division  at  the 
same  time. 

Ex.  1.   Multiply  6174  by  25. 

Since  25  =  100  h-  4,  we  shall  multiply  by  25  if  we  first  multiply 
by  100  and  then  divide  by  4.  For  by  multiplying  by  100  we  get  4 
times  too  much,  which  is  put  right  when  we  divide  by  4.  To  mul- 
tiply by  25  we  may  therefore  affix  two  naughts  and  divide  by  4 ; 
thus, 

4)617400 
154350 

Ex.  2.   Multiply  6174  by  125. 

Since  125  =  1000  +  8,  we  multiply  by  1000  and  then  divide  by  8, 
that  is,  we  affix  three  naughts  and  divide  by  8  ;  thus, 

8)6174000 
771750 

The  methods  adopted  in  the  following  examples  are  also  worth 
notice. 

Ex.  3.  Multiply  7964  by  9998. 

Since  9998  =  10000  -  2,  we  can  multiply  by  10000  and  by  2,  and 
take  the  difference  of  these  products. 

7964 
9998 


79640000 
15928 


79624072 
Ex.  4.  Multiply  7.964  by  9998. 
7.964 


79640. 

15.928 
79624.072 

72.  To  test  the  answer  in  division,  multiply  the  quo- 
tient by  the  divisor  (not  divisor  by  quotient),  and  to  the 
product  add  the  remainder  (if  any) ;  the  result  should 
equal  the  dividend.     [Art.  58.] 


Arts.  71-73.] 


EXAMPLES. 


51 


73.  Some  saving  of  time  in  division  will  be  effected  by 
performing  the  multiplication  of  the  divisor  and  the  sub- 
traction from  the  dividend  simultaneously ;  this  method 
should,  however,  be  attempted  only  by  those  who  show 
some  aptitude  for  numerical  calculations,  for  the  slight 
gain  in  speed  by  no  means  makes  up  for  the  increased 
liability  to  error. 

The  method  will  be  understood  from  the  following 
example : 

Divide  102739  by  29. 

29)102739(3542 
157 
123 
79 

21  rem. 
Explanation.  Instead  of  multiplying  29  by  3  and  subtracting 
the  whole  product  from  102,  we  subtract  the  several  figures  of  the 
product  as  we  go  along.  Thus,  3  times  9  are  27,  and  7  from  12 
leaves  5 ;  we  write  5,  and  carry  3  (2  from  the  27,  and  1  from  the 
12).  Then,  3  times  2  are  6,  and  3  (carried)  are  9,  and  9  from  10 
leaves  1.  The  remainder  is  15,  which  with  the  7  of  the  dividend 
makes  157  for  the  next  partial  dividend.     And  so  on  to  the  end. 

EXAMPLES    XIV. 
Written  Exercises. 


Divide 

1.  182  by  13.     4.  399  by  19.      7.  702  by  26. 

2.  204  by  17.     5.  575  by  23.      8.  1054  by  34. 

3.  221  by  17.     6.  899  by  29.      9.  4185  by  31. 

10.  1591  by  37.         14.  430686  by  71. 

11.  6016  by  94.         15.  415242  by  59. 

12.  710007  by  87.        16.  426713  by  47. 

13.  435435  by  65.  17.  562171  by  53. 


52  DIVISION.  [Chap.  II. 

18.  850902  by  78.  26.  21112  by  104. 

19.  1173021  by  97.  27.  185745  by  305. 

20.  1034550  by  95.  28.  801738  by  567. 

21.  2706420  by  86.  29.  8035370  by  2674. 

22.  11336  by  109.  30.  9570744  by  1593. 

23.  22563  by  207.  31.  407514744  by  6724. 

24.  160335  by  315.  32.  31587678  by  5067. 

25.  -39483  by  123.      33.  266  +  126  +  210  by  14. 

34.  6164  +  5226  by  67. 

35.  The  trees  in  an  orchard  are  arranged  in  153  rows, 
with  the  same  number  of  trees  in  each  row,  and  there 
are  16371  trees  altogether.  How  .many  trees  are  there 
in  each  row  ? 

36.  There  are  86400  seconds  in  a  day;  in  how  many 
days  are  there  13564800  seconds  ? 

b. 

In  division  of  decimals,  the  quotient  should  be  continued  until 
there  is  no  remainder,  unless  otherwise  directed.  This  can  be 
accomplished  by  annexing  naughts  to  the  dividend,  as  in  Ex.  6, 
Art.  66.  In  general  practice  three  or  four  decimal  places  in  the 
quotient  are  considered  sufficient. 

Divide 

1.  16.4  by  2.  3.    17.2  by  4.  5.    .288  by  9. 

2.  32.7  by  3.  4.    .156  by  6.  6.    .135  by  9. 

7.  125.6  by  20.  13.    5.22  by  48. 

8.  31.83  by  30.  14.    .171  by  72. 

9.  11.7215  by  50.  15.    .012  by  1600. 

10.  215.4  by  80.  16.    .027  by  45. 

11.  .0321  by  60.  17.   2.355  by  75. 

12.  .174  by  120.  18.   2.715  by  48. 


Art.  73.] 


EXAMPLES. 


53 


19.  52.7  by  17. 

20.  43.7  by  23. 

21.  166.6  by  119. 

22.  3751.5  by  123. 

23.  3.7515  by  1230. 

24.  375.15  by  125. 

25.  37.515  by  1250. 

Find,  to  4  places  of  decimals, 

32.  12.15  -r- 148.  35. 

33.  2.374 -=-156.  36. 

34.  41.75-89.  37.    121  -f- 170. 

38.    Simplify  .026  x  .0493  -r-  221. 


26.  67.77  by  135. 

27.  .006777  by  1350. 

28.  1.036656  by  207. 

29.  .1036656  by  5008. 

30.  .001036656  by  2070. 

31.  .651714  by  3156. 


135  -=- 17. 
17  -=- 135. 


Divide 

1.  6.2  by  .01. 

2.  .347  by  .001. 

3.  12.3  by  .0001. 

4.  3.5  by  .5. 

5.  .75  by  .05. 

6.  1.25  by  .005. 

7.  62.5  by  2.5. 

8.  .625  by  .025. 

9.  625  by  .0025. 

10.  1.1  by  .125. 

11.  .019  by  1.25. 

12.  170  by  .00125. 

25.  Find,  to  4  places 
.0167  -*-  3.17. 


13.  1.5  by  2.4. 

14.  5.76  by  4.8. 

15.  8.1  by  .36. 

16.  159.1  by  3.7. 

17.  6.016  by  .94. 

18.  70.992  by  8.7. 

19  .435435  by  .0065. 

20.  430.686  by  .0071. 

21.  415.242  by  .0059. 

22.  .185745  by  3.05. 

23.  4.07514744  by  .006724. 

24.  .9570744  by  159.3. 

of  decimals,  43.21  -f- 123.4,  and 


54  DIVISION.  [Chap.  II. 

26.  Simplify  360  --  7.2  -=-  .16. 

27.  Simplify  .0441  --  .21  -h  .56. 

28.  Simplify  1.953  --  8.68  x  .035. 

29.  How  many  lengths  eacli  2.56  inches  are  there  in  a 
rod  120  inches  long ;  and  how  much  is  left  over  ? 

30.  How  many  packets  of  tea,  each  containing  1.85 
ounces,  can  be  made  up  out  of  a  chest  containing  2400 
ounces ;  and  how  much  is  left  over  ? 

d. 

Divide,  using  factors  not  greater  than  12, 

1.  396  by  18.     3.  625  by  25.     5.  8820  by  36. 

2.  816  by  24.     4.  3753  by  27     6.  15750  by  42. 

7.  1958528  by  64.      18.  21574  by  20,  40,  and  60. 

8.  59081805  by  81.     19.  123456  by  20,  30,  and  40. 

9.  13339728  by  108.     20.  158937  by  20,  50,  and  70. 

10.  10654069140  by  132.  21.  2167  by  30,  and  50. 

11.  316794  by  45.       22.  16819  by  30,  and  80. 

12.  7196243  by  35.      23,  17943  by  40,  and  60. 

13.  2106935  by  36.      24.  21985  by  50,  and  90. 

14.  9172143  by  72.      25.  217943  by  500. 

15.  22222222  by  99.     26.  712415  by  700. 

16.  123456789  by  132.    27.  217643  by  216. 

17.  32163  by  20, 30,  and 40.  28.  1234567  by  242. 

e. 

Multiply,  using  the  short  process, 

1.  74562  by  25.  4.   387.4  by  125. 

2.  4.162  by  25.  5.   79.624  by  99. 

3.  12678  by  125.  6.    1897  by  999. 


Art.  73.]  MISCELLANEOUS  EXAMPLES.  55 

7.  29075  by  998.  10.    .6003  by  12.5. 

8.  .79184  by  9999.  11.   786  by  250. 

9.  6729  by  12.5.  12.   34.65  by  .0125. 

EXAMPLES  XV. 
Miscellaneous  Examples,  Chapters  I  and  II. 

1.  Express  in  words  3015602,  and  in  figures  eleven 
million  five  hundred  thousand  two  hundred  fourteen. 

2.  Find  the  sum  of  30157,  12.468,  31947,  and  3.6539. 

3.  By  how  many  is  13018  greater  than  12997? 

4.  Multiply  8000  by  1250,  and  3200  by  12345. 

5.  How  many  times  can  317  be  subtracted  from  1389, 
and  what  is  the  remainder  ? 


6.  Express  MDCCCLXXIX  in  the  Arabic  notation, 
and  1449  by  means  of  Roman  numerals. 

7.  Find  1325 +  3016 +79 +  90167. 

8.  Find  316  - 179  +  257  -  89  - 185  +  398  -  485. 

9.  Multiply  1234  by  4321  and  9009  by  31562. 

10.  How  many  nineteens  are  there  in  five  thousand, 
and  how  many  are  over  ? 

11.  By  how  much  does  the  sum  of  3.72  and  10.015  fall 
short  of  the  sum  of  7.216  and  6.52  ? 


12.  Express  in  words  1632057  and  300416720J  500. 

13.  Subtract  the  sum  of  3158,  2016,  and  5143  from 

mil. 

14.  Multiply  the  difference  between  seventy-six  mil- 
lion seventy-six  and  four  hundred  forty  thousand  four 
hundred  forty,  by  eleven  hundred  fourteen. 


56  MISCELLANEOUS  EXAMPLES.  [Chap.  II. 

15.  A  farmer  has  197  sheep  and  three  times  as  many 
lambs.     How  many  sheep  and  lambs  has  he  altogether  ? 

16.  Find   by   short  divisions   how  many  thirty-fives 
there  are  in  31578,  and  how  many  are  over. 

17.  Add  31.057,  156.0083,  2.61759,  and  .008347. 

18.  Subtract  the  difference  between  3.14  and  1.0625 
from  the  sum  of  1.00172  and  2.127. 


19.  By  how  many  is  one  million  eight  thousand  nine 
hundred  seventy-four  less  than  two  million  eleven  hun- 
dred twelve  ? 

20.  Find  3142  - 1250  -  989  +  6217  -  3587  - 1924. 

21.  A  farmer  had  2000  bags  of  wheat.  He  sold  527 
bags  to  one  man  and  255  bags  to  each  of  three  others. 
How  many  bags  were  left  unsold  ? 

22.  How  many  letters  are  there  in  a  book  of  375 
pages,  each  page  of  which  contains  32  lines,  and  each  line 

45  letters? 

23.  Multiply  31.025  by  .032,  and  .0625  by  .00125. 

24.  By  what  number  must  59755  be  divided  in  order 
that  the  quotient  may  be  19  ? 

25.  Divide  7.0175  by  17.5,  and  7.5  by  .00625. 


26.  In  one  school  there  are  one  hundred  seventy-six 
boys  and  one  hundred  and  twelve  girls ;  and  in  another 
school  there  are  half  as  many  boys  and  twice  as  many 
girls.  How  many  scholars  altogether  are  there  in  the 
two  schools? 

27.  The  sum  of  two  numbers  is  317205  and  one  of 
them  is  185964 ;  what  is  the  other  ? 


Art.  73.]  MISCELLANEOUS   EXAMPLES.  57 

28.  A  farmer  sold  75  cattle  at  24  dollars  a  head  and 
bought  with  the  money  sheep  at  2  dollars  each.  How 
many  sheep  did  he  buy  ? 

29.  Divide  .04312  by  .0044,  and  9.0225  by  .225. 

30.  Divide  358  by  15  by  short  divisions. 

31.  What  is  the  least  number  which  must  be  added  to 
57914  in  order  that  the  sum  may  be  exactly  divisible  by 
315? 

32.  Divide  the  product  of  37.5  and  .1248  by  .005625. 


33.  Express  MDCCCXCIY  in  the  Arabic  notation,  and 
2875  by  means  of  Roman  numerals. 

34.  In  a  school  of  four  hundred  and  ninety  children 
there  are  two  hundred  and  seventy-six  girls.  How  many 
more  girls  than  boys  are  there  ? 

35.  In  a  train  there  are  37  cars  each  having  seats  for 
36  people,  and  there  are  375  passengers  in  the  train ;  how 
many  seats  are  empty? 

36.  Simplify  1.702  x  2.9015  -h  .0005803. 

37.  Divide  the  product  of  .0374  and  .0075  by  the  dif- 
ference between  .675  and  .6375. 

38.  Show  that  the  sum  of  the  squares  of  three  thousand 
nine,  and  four  thousand  twelve,  is  equal  to  the  square  of 
five  thousand  fifteen. 

39.  What  is  the  least  number  which  must  be  subtracted 
from  2146537  in  order  that  the  remainder  may  be  exactly 
divisible  by  4275  ? 


40.  Subtract  nine  hundred  five  million  eight  thousand 
nine  hundred  sixty-five  from  eleven  hundred  million  two 
thousand  three  hundred,  and  express  the  result  in  words. 


58  MISCELLANEOUS  EXAMPLES.     [Chaps.  II. ,  III. 

41.  At  an  election,  the  successful  candidate,  who  ob- 
tained 12597  votes,  had  a  majority  of  1479  over  the  un- 
successful candidate.  How  many  votes  were  given 
altogether  ? 

42.  Find  2197-1982  +  374  +  10085-8216  +  11597 
-  7986. 

43.  Find  the  squares  of  2.15  and  .0324. 

44.  Multiply  16777216  by  131072,  also  divide  16777216 
by  131072,  and  express  the  results  in  words. 

45.  Find  the  least  number  of  repetitions  of  3745  whose 
sum  is  greater  than  a  million. 

46.  Divide  .378  by  262.5,  and  37.8  by  .02625. 


47.  Express  the  numbers  29,  47,  158,  679,  1464,  and 
10385  by  means  of  Roman  numerals. 

48.  How  many  figures  are  there  in  all  the  numbers 
from  1  to  100?  How  many  in  the  numbers  from  1  to 
1000? 

49.  A  certain  number  when  divided  by  3008  gives  a 
quotient  3875  and  a  remainder  2794.  What  is  the- 
number  ? 

50.  Divide  999999  by  the  continued  product  of  3,  7, 
11,  and  13. 

51.  The  sum  of  two  numbers  is  315642,  and  one  of  the 
numbers  is  twice  the  other :  find  them. 

52.  Divide  2722.05  by  .345,  and  .0272205  by  3.45. 

53.  Divide  (144.4  +  152x4.6)  by  19;  prove  your 
answer  by  dividing  after  uniting  the  terms  of  the 
dividend. 

54.  Divide,  by  factors,  (6.3  x  6  +  4.9  x  18)  by  21. 
See  Art.  44  for  definition  of  factors. 


Arts.  74,  75.]  FACTORS.  59 


CHAPTER  III. 

FACTORS  AND  MULTIPLES  —  SQUARE  ROOT  — HIGHEST 
COMMON  FACTOR  — LEAST  COMMON  MULTIPLE. 

Factors. 

74.  An  exact  divisor  of  a  number  is  called  a  Factor 
of  that  number ;  thus, 

2,  3,  4,  6,  and  12  are  factors  of  24.-    [Art.  44.] 

A  factor  is  also  called  a  Measure. 

A  number  that  is  exactly  divisible  by  another  number 
is  called  a  Multiple  of  that  number ;  thus, 

12,  30,  54,  72,  and  90  are  multiples  of  6. 

It  will  be  seen  at  once  that  a  number  has  a  limited  number  of 
factors,  but  an  unlimited  number  of  multiples. 

75.  A  number  which  is  not  divisible  by  any  number 
except  itself  and  1  is  called  a  Prime  Number,  or  a  Prime. 

Thus,  2,  3,  5,  7,  etc.,  are  primes. 

Every  number  which  has  other  factors  beside  itself 
and  unity  is  called  a  Composite  number. 
Thus,  4,  6,  8,  9,  etc.,  are  composite  numbers. 

Two  numbers,  both  of  which  cannot  be  divided  by  the 
same  number  (except  unity),  are  said  to  be  prime  to  one 
another. 

Thus,  4  and  9  are  prime  to  one  another;  both,  however,  are 
composite  numbers. 


60  FACTORS  AND   MULTIPLES.  [Chap.  III. 

76.  Numbers  divisible  by  2  are  called  Even  numbers. 
Numbers  not  divisible  by  2  are  called  Odd  numbers. 

2,  14,  30,  and  74  are  even  numbers. 

3,  7,  27,  and  51  are  odd  numbers. 

The  following  simple  conditions  of  divisibility  will  be 
found  to  be  useful : 

(i)  A  number  whose  last  digit  expresses  an  even  num- 
ber is  divisible  by  2. 

248  and  100694  are  divisible  by  2. 

(ii)  A  number  whose  last  digit  is  5  or  0  is  divisible 
by  5. 

25,  55,  and  600  are  divisible  by  5. 

(iii)  A  number  whose  last  two  digits  express  a  num- 
ber divisible  by  4  or  by  25  is  divisible  by  4  or  by  25, 
respectively. 

67215736  is  divisible  by  4. 
23798675  is  divisible  by  25. 

(iv)  A  number  the  sum  of  whose  digits  is  divisible  by 
3  or  by  9  is  divisible  by  3  or  by  9,  respectively. 

The  sum  of  the  digits  of  the  number  56174154,  namely, 

5  +  6  +  1  +  7  +  4  +  1+5  +  4,  is  33; 

and  33  is  divisible  by  3,  but  is  not  divisible  by  9.    Thus,  the 
number  56174154  is  divisible  by  3,  but  not  by  9. 

(v)  A  number  is  divisible  by  11  when  the  difference 
between  the  sum  of  the  first,  third,  fifth,  etc.,  digits  and 
the  sum  of  the  second,  fourth,  sixth,  etc.,  digits  is  zero 
or  a  multiple  of  11,  and  not  otherwise. 

Thus,  3572129  is  seen  to  be  divisible  by  11,  since  9  +  1  +  7  +  3 
differs  from  2  +  2  +  5  by  11. 


Arts.  76,  77.]  EXAMPLES.  61 

EXAMPLES    XVI. 
Oral  Exercises. 

Which  of  the  numbers,  2,  4,  8,  3,  9,  5,  25,  125,  11,  can 
be  seen  by  inspection  to  be  factors  of 

1.  964.  4.    7326.         7.   94680.         10.   49125. 

2.  225.  5.   6975.         8.   29304.         11.   307890. 

3.  1925.         6.    4125.         9.    76164.         12.    264792. 

77.   The  following  are  important  general  theorems : 

I.  Every  divisor  or  factor  of  each  of  several  numbers  is 
a  divisor  of  their  sum. 

If,  for  example,  each  of  several  numbers  is  divisible  by  12,  then 
each  can  be  arranged  in  groups  of  twelve,  and  therefore  their  sum 
consists  of  a  certain  number  of  twelves.  Similarly  for  any  other 
divisor. 

II.  Every  divisor  of  a  number  is  a  divisor  of  any  mul- 
tiple of  that  number. 

If,  for  example,  any  number  is  divisible  by  12,  it  can  be  arranged 
in  groups  of  twelves,  and  so  also  can  any  number  of  repetitions  of 
the  number. 

III.  Every  divisor  of  two  numbers  is  a  divisor  of  the 
sum,  or  of  the  difference,  of  any  multiples  of  the  numbers. 

If,  for  example,  two  numbers  are  both  divisible  by  12,  they  can 
both  be  arranged  in  groups  of  twelves,  and  so  also  can  any  multiples 
of  either.  These  multiples  can  then  be  added,  or  one  can  be  taken 
from  the  other,  without  taking  to  pieces  any  of  the  groups. 

To  make  the  above  theorems  quite  clear  to  a  beginner, 
it  would  be  well  to  have  actual  counters  to  deal  with, 
which  could  be  tied  up  by  twelves  in  bags  or  bundles. 
The  pupil  would  then  see  that  the  different  additions 
and  subtractions  could  be  performed  without  undoing  any 


62  FACTOKS  AND   MULTIPLES.  [Chap.  III. 

of  the  bags  or  bundles,  and  therefore  the  final  resnlt  must 
be  a  certain  number  of  twelves. 

78.  The  Sieve  of  Eratosthenes.  —  The  different  prime 
numbers  can  be  found  in  order  by  the  following  method, 
called  the  Sieve  of  Eratosthenes. 

Write  in  their  natural  order  the  numbers  from  1  to  any  extent 
that  may  be  required  ;  thus, 

1,  2,  3,  4,  5,  6,  7,  8,  9,  i6, 

11,  12,  13,  i4,  i5,  16,  17,  i8,  19,  20, 

2i,  22,  23,  24,  25,  26,  27,  28,  29,  30, 

31,  32,  33,  34,  35,  36,  37,  38,  39,  40,  etc. 

Now  take  the  first  prime  number,  2,  and  over  every  second  number 
from  2  place  a  dot :  we  thus  mark  all  the  multiples  of  2.  Then, 
leaving  3  unmarked,  place  a  dot  over  every  third  number  from  3  :  we 
thus  mark  all  multiples  of  3.  The  number  next  to  3  left  unmarked 
is  5  ;  and,  leaving  5  unmarked,  place  a  dot  over  every  fifth  number 
from  5 :  we  thus  mark  all  multiples  of  5.  And  so  for  multiples 
of  7,  etc. 

By  proceeding  in  this  way  all  multiples  of  the  prime  numbers, 
2, 3,  5, 7,  etc.,  are  struck  out ;  also  multiples  of  all  composite  numbers 
are  necessarily  struck  out  at  the  same  time  :  for  example,  all  multi- 
ples of  6  are  struck  out  as  being  multiples  of  either  of  its  prime 
factors  2  or  3.  Hence  all  the  numbers  which  are  left  unmarked  are 
primes,  for  no  one  of  them  is  divisible  by  any  number  (except  unity) 
which  is  smaller  than  itself. 

We  can  thus  find  in  order  as  many  prime  numbers  as  we  please. 

The  primes  less  than  100  will  be  found  to  be 

1,     2,     3,     5,     7,  11,  13,  17,  19,  23,  29,  31,  37,  41, 
43,  47,  53,  59,  61,  67,  71,  73,  79,  83,  89,  and  97. 

79.  To  find  whether  a  given  number  is  or  is  not  a  prime, 
we  have  only  to  see  whether  it  is  divisible  by  any  one  of 
the  prime  numbers,  2,  3,  5,  7,  etc. 


Arts.  78-80.] 


PRIME   FACTORS. 


63 


Ex.  1.    Is  233  a  prime  number  f 

By  trial  it  will  be  found  that  233  is  not  divisible  by  2,  nor  by  3, 
nor  by  5,  nor  by  7,  nor  by  11,  nor  by  13,  nor  by  17.  Now  it  is  not 
necessary  to  try  any  other  primes,  for  233  -^-17  gives  a  quotient  less 
than  17  ;  if,  therefore,  233  were  divisible  by  a  prime  greater  than  17, 
the  quotient  would  be  less  than  17,  and  233  would  be  divisible  by 
this  quotient,  that  is  by  a  number  less  than  17,  which  we  know  is 
not  the  case.    Hence  233  is  a  prime  number. 


80.   Resolution    into    Prime    Factors. 

The  following  examples  will  suffice  to  show  how  to 
express  any  number  whatever  as  the  product  of  factors 
each  of  which  is  a  prime. 

The  method  is  applicable  to  all  numbers  however  large,  provided 
we  find  as  many  prime  numbers  as  may  be  necessary  by  means 
of  the  ■  sieve ' ;  the  method  would,  however,  be  extremely  tedious 
in  the  case  of  a  very  large  number. 


Ex.  1.   Express  28028  as  the  product  of  prime  factors. 

28028  =  2  x  14014 

=  2  x  2  x  7007 
=  2  x  2  x  7  x  1001 
=  2x2x7x7x143 
=  2x2x7x7x11x13. 


These  continuous  divisions  may 
be  thus  expressed : 
28028 


14014 


7007 


1001 


143 


18 


Ex.  2.   Find  the  prime  factors  of  3978. 


2 

3978 

3 

1989 

3 

663 

13 

221 

17 
The  answer  is  2,  32,  13,  and  17. 

Ex.  3.    Obtain  two  factors  of  14  +  22. 

14  +  22  =  2  multiplied  by  (7  +  11). 


64  FACTORS  AND   MULTIPLES.  [Chap.  III. 

EXAMPLES   XVII. 

Express  the  following  numbers  as  products  of  prime 
factors : 

Oral  Exercises. 

1.  6,  9,  10,  15,  24,  30,  36,  39,  45,  48. 

2.  .6,  .9,  1.5,  2.4,  3.6,  3.9,  4.5,  4.8. 

3.  .09,  .15,  .24,  .36,  .39,  .45,  .48. 

4.  49,  50,  54,  60,  5.4,  75,  81. 

5.  3.2,  100,  120,  130. 

"Written  Exercises. 

6.  184,  196,  275,  273,  391,  525. 

7.  350,  459,  715,  728,  792,  999. 

8.  1092,3885. 

9.  51051,  74613,  462462. 

10.  Obtain  two  factors  of  (6  +  15). 

11.  Obtain  three  factors  of  (30  +  70). 

12.  Obtain  two  factors  of  (2x6  +  4x5  +  2  x  17). 

Square  Root. 

81.  Obtain  the  two  equal  factors  of  4 ;  of  9 ;  of  25 ; 
of  0.4. 

Obtain  the  three  equal  factors  of  8 ;  of  27 ;  of  .008. 

Obtain  the  four  equal  factors  of  16  ;  of  81. 

One  of  the  equal  factors  of  a  number  is  called  a  Root 
of  the  number ;  thus,  3  is  a  root  of  9 ;  5  is  a  root  of  25 ; 
3  is  a  root  of  27 ;  .2  is  a  root  of  .04 ;  .2  is  a  root  of  .008. 

If  a  number  is  the  product  of  tivo  equal  factors,  its  root 
is  called  a  second  root,  or  Square  Root. 

If  a  number  is  the  product  of  three  equal  factors,  its 
root  is  called  a  third  root,  or  Cube  Root. 

Likewise  we  have  fourth  smdjifth  roots,  etc. 


Arts.  81-85.]  SQUARE   ROOT.  6$ 

82.  It  was  shown  in  Art.  52  that  a  square  is  obtained 
when  the  multiplicand  equals  the  multiplier. 

Here  it  is  seen  that  a  square  root  is  obtained  when  the 
quotient  equals  the  divisor. 

83.  The  squares  of  the  first  12  whole  numbers  should 
be  known :  they  are 

1,  4,  9,  16,  25,  36,  49,  64,  81,  100,  121,  144. 

It  will  be  seen  at  once  that  the  square  root  of  an  integer  is  by 
no  means  always  an  integer ;  in  fact  the  only  numbers  between  1 
and  100  which  have  an  integral  square  root  are  4,  9,  16,  25,  36,  49, 
64,  and  81. 

It  will  be  seen  later  on  that  the  square  root  of  an  integer  which 
is  not  the  square  of  a  whole  number  can  be  found  approximately* 
only. 

An  integer  (or  a  decimal)  which  is  the  square  of 
another  integer  (or  decimal)  is  called  a  Perfect  Square. 

Thus,  16  and  .09  are  perfect  squares  ;  namely,  the  squares  of  4 
and  .3,  respectively. 

84.  The  sign  ^/  is  used  to  indicate  a  root,  and  is  called 
the  Radical  Sign. 

If  any  other  root  than  the  second  is  to  be  indicated, 
a  small  figure  called  an  Index  is  placed  just  above  the 
radical  sign ;  thus, 

-y/9  indicates  the  square  root  of  9 ; 
■y/8  indicates  the  cube  root  of  8 ; 
.J/243  indicates  the  fifth  root  of  243. 

85.  In  simple  cases,  the  square  root  of  a  given  number 
can  be  found  by  separating  it  into  factors  which  are 
squares,  and  making  use  of  the  principle  that  the  product 
of  the  squares  of  two  or  more  quantities  is  equal  to  the 
square  of  the  product  of  those  quantities. 


66  FACTORS  AND   MULTIPLES.  [Chap.  III. 

For  example,  to  find  V324- 

324  =  4  x  81         =  22  x  92  =  (2  x  9)2  ; 
hence,         V324  =  V(2  x  9)2  =  2  x  9    =18. 

Also,      V1-44  =  V(22  x  -62)  =  V(2  x  .6)2  =  2  x  .6  =  1.2. 

EXAMPLES    XVIII. 
Written  Exercises. 

Find  the  square  roots  of  the  following  numbers : 

1.  196;  1.96.        6.    576;  5.76.        11.   2601. 

2.  225;  2.25.        7.    676;  6.76.        12.    3969;  .003969. 

3.  324.  8.    1089;. 1089.     13.   4225;  42.25. 

4.  400;  4.84.        9.    1225;  12.25.     14.    7056. 

5.  441;  4.41.      10.   2025.  15.    11025. 

In  each  of  the  following  numbers,  what  is  the  least 
multiplier  that  will  produce  a  perfect  square  ? 

16.  12.  18.    24.  20.    126.  22.    1176. 

17.  20.  19.    52.  21.    140.  23.    1344. 
24.    State  a  number  which  has  a  second  and  a  fourth 

root ;  a  second,  third,  and  sixth  root. 

86.  The  above  method  cannot  be  easily  used  in  all 
cases,  but  the  method  which  can  be  used  will  be  under- 
stood from  the  following  explanation.     [Arts.  86,  87,  88.] 

Let  it  be  required  to  find  632.  This  may  be  done  in 
the  usual  way,  and  the  square  is  found  to  be  3969. 

Now  632  may  be  written  (60  +  3)2,  which  equals  the  square  of 
60  +  twice  the  product  of  60  by  3  +  the  square  of  3. 

60  +  3 
60  +  3 
60  x  3  +  32 
602  +  60  x  3 


602  +  2(60x3)+32 


Arts.  86-88.]  SQUARE   ROOT.  67 

The  square  of  the  sum  of  any  other  pair  of  numbers 
can  be  expressed  in  a  similar  form. 

Hence,  the  square  of  the  sum  of  any  two  numbers  is  equal 
to  the  sum  of  their  squares  plus  twice  their  product. 

87.  Since 

.012=    .0001,  102=         100, 

.l2    =    .01,  1002=      10000, 

l2.       =  1,  10002  =  1000000, 

and  so  on,  it  follows  that  if  a  number  has  one  digit,  its 
square  has  either  one  or  two  digits ;  if  a  number  has  two 
digits,  its  square  has  either  three  or  four  digits;  if  a 
number  has  three  digits,  its  square  has  either  five  or  six 
digits ;  and  so  on. 

Hence,  if  we  mark  off  the  digits  of  a  given  number, 
beginning  at  the  units'  digit,  into  periods  of  two,  the  last 
of  the  periods  on  the  left  containing  either  one  or  two 
digits ;  then  the  number  of  these  periods  will  be  equal  to 
the  number  of  digits  in  the  square  root  of  the  given  number. 

For  example,  by  pointing  off  the  numbers,  961,  54.76,  36.8449, 
1522756,  thus,  9'61, 54'.76,  36'.84'49, 1'52'27'56,  we  see  that  the  square 
roots  of  these  numbers  contain  2,  2,  3,  and  4  figures,  respectively. 

88.  To  find  the  Square  Root  of  Any  Number. 

The  method  will  be  seen  from  the  following  examples : 

Ex.  1.     To  find  the  square  root  of  3969. 

39'69'  f  60  +  3       By  pointing  off  the  digits  into  Periods 

36  00  °^  two'  we  see  tliat  tliere  are  two  digits 

2 x60+3  —  123Y369"  *n  tne  re(luired  root ;  and,  since  602  = 

3  69  3600  and  702  =  4900,  we  see  that  the 

root  lies  between  60  and  70.    The  tens' 

digit  must  therefore  be  6,  and  we  have  now  to  find  the  units'  digit. 

If  we  subtract  602  from  the  given  number,  the  remainder  is  369  ; 

and,  by  Art.  86,  this  remainder  is  equal  to  (2  x  60)  times  units' 

digit  +  (units'  digit)2,  or  units'  digit  times  (2  x60  -f  units'  digit)  ; 


68  FACTORS  AND   MULTIPLES.  [Chap.  III. 

i.  e. ,  369  is  the  product  of  the  unknown  digit  by  (2  x  60  +  the 

unknown  digit). 

Hence,  if  we  use  2  x  60  as  a  trial  divisor,  we  obtain  a  quotient, 

namely  3,  which  is  either  equal  to  or  greater  than  the  required  digit. 

Put  this  quotient  for  the  unknown  digit,  and  we  have  (2  x  60  +  3), 

or  123,  as  a  true,  or  complete,  divisor.     Now  dividing  369  by  123, 

we  find  that  3  is  the  correct  digit  for  units'  place. 

39'69'f63  The  Process  is  shortened,  as  in  ordinary  division, 

3g  by  the  omission  of  zeros ;  the  periods,  of  two  figures 

123)369  each,  are  brought  down  one  at  a  time,  one  figure  of 

369  the  root  corresponding  to  each  period. 

Ex.  2.     Find  the  square  root  of  114244. 

11'42'44'(300  +30  +  8  11 '42 '44  (338 

9  00  00  _9 

600  +  30)2  42  44  63)2  42 

1  89  00  1  89 


660  +  8)  53  44  668)  53  44 

53  44  53  44 

There  are  here  three  periods  and  therefore  three  digits  in  the  root, 
the  first  of  which  is  3,  since  114244  is  between  3002  and  4002.  Using 
300  x  2  as  a  trial  divisor  in  order  to  find  the  second  figure  in  the  root, 
we  obtain  the  quotient  40 ;  this,  however,  is  too  great,  for  (600  +  40), 
the  complete  divisor,  is  not  contained  40  times  in  the  dividend  ;  we 
therefore  try  30,  which  proves  to  be  correct. 

The  process  is  usually  indicated  in  the  shortened  form,  any 
trial  divisor  being  the  product  of  the  quotient  already  found  by 
2  and  10  continuously,  while  the  corresponding  complete  divisor  is 
the  trial  divisor  with  its  naught  displaced  by  the  quotient  figure 
obtained  in  using  the-  trial  divisor :  thus,  in  Ex.  2,  the  first  trial 
divisor  is  3  x  2  x  10  =  60,  while  the  complete  divisor  is  63 ;  also 
the  second  trial  divisor  is  33  x  2  x  10  =  660,  while  the  complete 
divisor  is  668. 

Ex.  3.   Find  the  square  root  of  50126400. 

Here  there  are  four  periods  and  therefore 
50'12'64'00'(7080    f0Ur  figures  in  the  root.     A  figure  of  the 
root  corresponds  to  each  period  brought 


1408)1 12  ^  down  in  the  shortened  process ;  and  in  the 

^  present  case  two  figures  of  the  root  are 

naughts. 


Arts.  88,  89.]  SQUARE   ROOT.  69 

Ex.  4.    Find  y/U. 44. 

In  the  case  of  a  decimal,  the  pointing  must  be  begun  at  the 
decimal  point,  and  carried  to  the  left  for  the  integral  part,  and  to 
the  right  for  the  decimal  part. 

14.'44'(3.8 
9 


68)5  44 
5  44 


Ex.  5.    Find  y/SVo. 


Having  used  both  periods  of  the  given 
3'15.'00'00'(17.74  +     number,  there  is  a  remainder  of  26.     We 

_J place  a  decimal  point  after  the  units'  figure 

27)2  15  0f  both  dividend  and  quotient,  and  then 

m continue  the  periods  by  using  naughts. 

'    94  2Q  ^ke  process  would  never  terminate,  hence 

or.A±\  i  7i  on  315  is  not  a  perfect  square.     We  obtain, 

X  41  76  however,  an  approximate  answer  by  stop- 

29  24  ping  after  the  second  or  third  decimal 

place. 

89.  Since  the  square  of  a  number  cannot  end  with  a  naught 
unless  the  number  itself  ends  with  a  naught,  it  follows  that,  if  the 
process  of  finding  a  square  root  does  not  terminate  when  the 
last  significant  figure  is  brought  down,  the  process  will  never 
terminate. 

Expressions  such  as  -^/S,  y/2.5,  which  cannot  be  found  exactly  are 
called  Surds. 

Although  no  definite  number  can  be  found  whose  square  is  exactly 
equal  to  3,  the  process  of  Art.  88,  Ex.  5,  if  continued  far  enough, 
will  enable  us  to  find  a  decimal  whose  square  differs  from  3  by  as 
small  a  quantity  as  we  please. 

EXAMPLES  XIX. 
Written  Exercises. 

Find  the  square  roots  of 

1.  729.  4.   .1849.  7.  16.81. 

2.  3481.         5.  2209.  8.  56169. 

3.  11.56.         6.  6084.  9.  4157521. 


70  FACTORS  AND   MULTIPLES.  [Chap.  III. 

10.  49126081.  13.   9345249.  16.   13.69. 

11.  26625600.  14.   934.5249.  17.   136.9. 

12.  182.493081.    15.  1369.        18.  1.369. 

19.  .00022201.  20.  2.2201. 

Find,  to  three  decimal  places, 

21.  V5-  24«    V125-4-  27-  V-081. 

22.  V19-  25-    V31-046-  28-  V-01735- 

23.  V21-5-  26-    V-4-  29-  V-0002- 

Highest  Common  Factor. 

90.  A  number  which  exactly  divides  two  or  more  num- 
bers is  called  their  Common  Factor. 

For  example,  2,  3,  and  6  are  common  factors  of  18  and  24. 

The  largest  number  which  exactly  divides  two  or  more 
numbers  is  called  their  Highest  Common  Factor  (H.C.F.)  ; 
called,  also,  the  Greatest  Common  Measure  (G.C.M.),  and 
the  Greatest  Common  Divisor  (G.C.D.). 

Thus,  6  is  the  H.C.F.  of  18  and  24, 
or  the  G.C.M.  of  18  and  24, 
or  the  G.C.D.  of  18  and  24. 

91.  After  numbers  have  been  resolved  into  their  prime 
factors,  their  H.C.F.  can  be  found  by  inspection. 

Consider,  for  example,  the  numbers  30  and  42. 
30  =  2  x  3  x  5 

42  —  2  x  3  x  7         Here  we  see  that  2  an(*  '  are  tlie  on^y 

primes  that  are  divisors  of  both  30  and  42. 

H.C.F.  =2x3  Therefore  the  H.C.F.  =2x3  =  6. 
=  6. 

Again, 

720  =  2*  x  32  x  5 

1080  =  23  x  38  x  5         Here  2  is  a  common  factor  three  times, 
H.C.F.  =  23  x  32  x  5    3  is  common  twice,  and  5  is  common  once. 
=  360. 


Arts.  90-92.]      HIGHEST   COMMON  FACTOR.  71 

The  H.C.F.  of  two  or  more  numbers  must  be  the 
continued  product  of  all  the  common  prime  factors  of  the 
numbers. 

Further  Illustrations. 

Ex.  1.  Ex.  2. 

792  =  23  x  32  x  11  2730  =  2    x3x5x7xl3 

4368  =  2*  x  3   x    7  x  13        5304  =  23x3  x  13  x  17 

H.C.F.  =  23x3  780  =  22  x  3  x  5         x  13 

=  24.  H.C.F.=  2    x3  xl3 

=  78. 

EXAMPLES  XX. 
Oral  Exercises. 

Find  the  H.C.F.  of   ' 

1.  12  and  18.      3.  30  and  42.      5.  60  and  84. 

2.  20  and  25.      4.  18  and  30.      6.  54  and  90. 

"Written  Exercises. 

7.  45  and  105.        10.   189  and  273.      13.    693  and  819. 

8.  72  and  90.  11.    132  and  252.      14.    792  and  924. 

9.  126  and  315.      12.   315  and  357.      15.   891  and  1221. 

16.  48,  60,  and  72.  18.   264,  360,  and  600. 

17.  72,  108,  and  180.  19.    630,  756,  and  1155. 

92.  We  must  now  show  how  to  find  the  H.C.F.  of  two 
numbers  without  going  through  the  troublesome  process 
of  expressing  the  numbers  as  the  product  of  prime 
factors. 

The  method  depends  on  the  following  theorem,  proved 
in  Art.  77 : 

Any  common  factor  of  two  numbers  is  also  a  factor  of 
the  sum,  or  of  the  difference,  of  any  multiples  of  the 
numbers. 


72  FACTORS  AND  MULTIPLES.  [Chap.  III. 

Suppose  that  we  have  two  numbers  whose  H.C.F.  is 
required. 

If  we  divide  the  greater  number  by  the  smaller,  then, 
by  the  nature  of  division, 

(i)  the  remainder  is  equal  to  the  difference  between 
the  greater  number  and  some  multiple  of  the  smaller ; 

(ii)  the  greater  number  is  equal  to  the  sum  of  the 
remainder  and  some  multiple  of  the  smaller. 

From  (i)  it  follows  that  any  common  factor  of  the 
original  numbers  is  a  factor  of  the  remainder,  and  there- 
fore is  a  common  factor  of  the  remainder  and  the  smaller 
number. 

From  (ii)  it  follows  that  any  common  factor  of  the 
remainder  and  the  smaller  number  is  a  factor  of  the 
greater  number  also,  and  therefore  is  a  common  factor  of 
the  two  original  numbers. 

The  H.C.F.  of  the  two  original  numbers  must  therefore 
be  the  same  as  the  H.C.F.  of  the  smaller  number  and  the 
remainder. 

Thus  the  problem  of  finding  the  H.C.F.  of  the  two 
original  numbers  is  reduced  to  that  of  finding  the  H.C.F. 
of  the  smaller  number  and  the  remainder. 

Ex.  1.    Find  the  H.C.F.  o/3663  and  5439. 

Divide  the  greater  by  the  less. 

3663)5439(1 
3663 
1776 

Hence  the  H.C.F.  required  is  the  same  as  the  H.C.F.  of  1776 
and  3663.     Divide  the  greater  of  these  by  the  less. 

1776)3663(2 
3552 
111 


Art.  02.]  HIGHEST  COMMON  FACTOR.  73 

The  problem  is  now  reduced  to  finding  the  H.C.F.  of  111  and 

1776.    Again  divide. 

111)1776(16 
111 
666 
666 

Thus,  111  is  a  factor  of  1776,  and  therefore  111  is  the  H.C.F. 
of  111  and  1776. 

But  the  H.C.F.  of  111  and  1776  is  the  H.C.F.  required. 

The  successive  divisions  are  usually  written  in  a  more  compact 

form,  as  follows  : 

3663)5439(1 
3663 

1776)3663(2 
3552 
111)1776(16 
111 
666 
666 

Ex.  2.  Find  the  H.  C.  F.  of  311  and  331. 

311)331(1 
311 

20)311(15 

20_ 

111 

100 

11)20(1 

11 

9)11(1 
9 

2)9(4 
8 

1)2(2 
2 

Here  the  H.C.F.  of  311  and  331  is  the  same  as  the  H.C.F.  of 

1  and  2,  so  that  the  numbers  are  prime  to  one  another. 

In  this  example,  it  would  be  a  great  waste  of  time  to  proceed  to 
the  end;  for  the  H.C.F.  required  is  the  H.C.F.  of  any  divisor  and 
the  corresponding  dividend,  and  as  soon  as  it  is  obvious  that  one 
such  pair  have  no  common  factors  it  is  not  necessary  to  proceed 
further.  Now  the  only  prime  factors  of  20  are  2  and  5,  and  by 
inspection  neither  of  these  is  a  factor  of  311. 


1.080  = 

:2*X 

33  x 

5  x 

.001 

.072  = 

23  X 

32 

X 

.001 

H.C.F.  = 

23X 

32 

X 

.001 

.072, 

Find  the  H.C.F. 

of  .108  and  .072. 

.108  = 

22x 

33  x 

.001 

.072  = 

23  x 

32  x 

.001 

H.C.F.  = 

22  x 

32  x 

.001 

— 

.036. 

74  FACTORS  AND   MULTIPLES.  [Chap.  III. 

93.  The  H.C.F.  of  numbers  containing  decimals  is  not 
often  needed.  The  process,  however,  for  finding  such  is 
as  follows : 

Arrange  all  the  numbers,  by  annexing  naughts,  so  that 
all  shall  have  the  same  number  of  decimal  places ;  then 
proceed  as  before. 

Ex.  1.     Find  the  H.  C.F.  of  1.08  and  .072. 


Ex.  2. 


Ex.  3.     Find  the  H.  C.F.  of  366.3  and  54.39. 

366.30  =  2  x  32  x  5   x  11  x  37  x  .01 
54.39  3    x  72  x  37  x  .01 

H.C.F.  =        3    x  37  x  .01 
=  1.11. 

EXAMPLES  XXI. 
Written  Exercises. 

Find  the  H.C.F.  (or  G.C.M.)  of 

1.  221  and  247.  8.  4899  and  5893. 

2.  357  and  391.  9.  9709  and  22849. 

3.  899  and  1073.  10.  11663  and  12091. 

4.  663  and  923.  11.  17947  and  29737. 

5.  1517  and  1927.  12.  11453  and  12961. 

6.  1785  and  2485.  13.  3834038  and  4169594. 

7.  3499  and  3953.  14.  132038  and  369792. 


15.  5411728  and  10902416. 


Arts.  93-95.]      LEAST   COMMON  MULTIPLE.  75 

To  find  the  H.C.F.  of  three  or  more  numbers,  we  have 
only  to  find  the  H.C.F.  of  the  first  two  numbers ;  then  the 
H.C.F.  of  this  result  and  the  third  number ;  and  so  on. 

Ex.     Find  the  H.  C.F.  of  286,  338,  and  585. 
The  H.C.F.  of  286  and  338  is  26.     Then  the  H.C.F.  of  26  and 
585  is  13. 

Find  the  H.C.F.  of 

16.  165, 198,  242.  19.  2387,  2821,  4433. 

17.  312,429,572.  20.  3157,3321,4059. 

18.  222,370,550.  21.  4732,5824,6643. 

22.  What  would  be  the  answers  to  7,  9,  and  18,  if 
they  were  as  follows  ? 

(7)  3.499  and  395.3. 

(9)  97.09  and  2.2849. 

(18)  2.22,  370,  and  550. 

Least  Common  Multiple. 

94.  A  number  which  is  exactly  divisible  by  two  or 
more  numbers  is  called  a  Common  Multiple  of  those 
numbers. 

For  example,  200  is  a  CM.  of  20  and  25. 

The  smallest  number  which  is  exactly  divisible  by  two 
or  more  numbers  is  called  the  Least  Common  Multiple 
(L.C.M.)  of  those  numbers. 

For  example,  100  is  the  L.C.M.  of  20  and  25. 

95.  When  numbers  are  resolved  into  their  prime  fac- 
tors, their  L.C.M.  can  be  found  by  inspection: 

Consider,  for  example,  the  numbers  120,  252,  and  3575. 
120  =  23  x  3   x  5 
252  =  22  x  3*  x  7 
4125  =  _      3    x  53  x  11 
L.C.M.  =  23  x  32  x  53  x  7  x  11 
=  693000. 


76  FACTOKS   AND  MULTIPLES.  [Chap.  III. 

Here  we  see  that  a  common  multiple  must  contain  the  prime 
factors  2,  3,  5,  7,  and  11.  Also  a  multiple  of  the  first  number  must 
contain  the  third  power  at  least  of  the  factor  2  ;  a  multiple  of  the 
second  must  contain  the  second  power  at  least  of  the  factor  3 ;  a 
multiple  of  the  third  number  must  contain  the  third  power  at  least 
of  the  factor  5 ;  and  we  must  also  have  the  first  power  at  least  of 
the  factors  7  and  11. 

The  least  common  multiple  must  therefore  be 

2x2x2x3x3x5x5  x5x7x  11. 

The  L.C.M.  of  two  or  more  numbers  must  be  the  con- 
tinued product  of  the  highest  powers  of  all  the  different 
prime  factors  of  the  numbers. 

105  =  3  x  5   x  7 
Illustration.  126  =  2   x  32  x  7 

196  =  22  x  72 


L.C.M.  =22  X32x  5  x  72 

=  8820. 

96.  The  form  for  finding  the  L.C.M.  given  in  Art.  95  is  ex- 
cellent, because  of  the  ease  with  which  it  may  be  applied  to  any 
example  in  which  the  H.C.F.  or  the  L.C.M.  is  to  be  found,  and 
because  it  involves  more  or  less  mental  work,  which  is  stimulating. 

In  obtaining  the  prime  factors,  the  highest  powers  of  the  lowest 
primes  should  always  be  taken  out  first;  thus,  2,  22,  23,  etc.,  then 
3,  32,  33,  etc.,  should  be  the  order  in  which  the  factors  should  be 
taken  out. 

The  usual  method  by  which,  the  L.C.M.  of  numbers  is 
found  is  as  follows  : 

Having  written  the  numbers  in  a  row,  first  strike  out  any  num- 
bers which  are  factors  of  any  of  the  others  {for  every  multiple  of 
a  number  is  a  multiple  of  any  factor  of  that  number).  Then 
divide  by  any  prime  which  divides  at  least  two  of  the  numbers,  and 
put  the  quotients  below  those  numbers  which  can  be  divided,  and 
bring  down  all  those  numbers  which  are  not  divisible  by  that  prime. 
Operate  on  the  second  row  in  the  same  maimer  as  on  the  first,  and 
go  on  until  a  row  is  arrived  at  in  which  no  two  numbers  have  any 


[    UNIVERSITY   j 

Art.  96.]  EXAMPLES.  77 

common  factor.     Then  the  L.C.M.  is  the  continued  product  of  all 
the  divisors  and  all  the  numbers  left  in  the  last  row. 

Thus,  to  find  the  L.C.M.  of  2,  5,  15,  24,  25,  30,  36. 

The  process  is  written  as  follows : 

2)2,  ft  m  24,  25,  30,  36 

2)12,  25,  15,  18 

3)  6,  25,  15,    9 

2,  25,    ft    3 

Hence,  the  L.C.M.  is2x2x3x2x25x3  =  1800. 

The  reason  the  L.C.M.  is  given  by  the  above  process  is  that  the 
L.C.M.  of  the  numbers  in  the  first  row  =  2  x  L.C.M.  of  the  num- 
bers in  the  second  row  ;  and  so  on  to  the  end. 


EXAMPLES    XXII. 
Oral  Exercises. 

Find  the  L.M.C.  of 

1.  4,10.  5.  9,12.  9.  30,50. 

2.  6,  8.  6.  3,  7,  10.  10.  10,  35. 

3.  5,  10,  12.  7.  12,  20.  11.  3,  4,  5,  6. 

4.  7,  6,  2.  8.  8,  12,  3.  12.  2,  8,  12,  20. 

Written  Exercises. 

Find  the  L.C.M.  of 

13.  12,16.  16.  8,12,20.  19.  30,45,54. 

14.  20,  25.  17.  6,  8,  12.  20.  10,  14,  35. 

15.  25,  30.  18.  15,  20,  30.  21.  6,  12,  18,  63. 

22.  9,  36,  45,  81.  25.    16,  20,  22,  33,  36. 

23.  3,  11,  18,  33,  36.  26.    3,  4,  10,  12,  14, 16, 18. 

24.  25,  27,  33,  55.  27.   3,  14,  12,  56,  and  28. 


78  FACTORS  AND   MULTIPLES.  [Chap.  III. 

28.  16,18,20,24,30,36. 

29.  12,  15,  18,  21,  25,  35,  210. 

30.  12,42,49,54. 

31.  30,35,42,60,72. 

32.  18,  54,  90,  102,  120,  144. 

97.  To  find  the  L.C.M.  of  numbers  which  cannot  be 
readily  resolved  into  factors,  we  must  first  find  the  H.C.F. 

For  example,  to  find  the  L.C.M.  of  4592  and  5371. 
The  H.C.F.  is  41.     And  by  division 

4592  =  41  x  112. 

5371  =  41  x  131. 
Hence,  as  112  and  131  are  prime  to  one  another,  the  L.C.M.  is 

41  xll2  x  131. 

Hence  the  L.C.M.  of  two  numbers  equals  one  of  the 
numbers  multiplied  by  the  quotient  obtained  by  dividing 
the  other  number  by  their  H.C.F. 

It  should  be  noticed  that  the  L.C.M.  x  G.C.M.  =  41  xll2  x  41 
X  131  =  product  of  the  numbers. 

To  find  the  L.C.M  of  more  than  two  numbers  we  can 
find  the  L.C.M.  of  two  of  the  numbers,  and  then  the 
L.C.M.  of  this  result  and  of  the  third  number,  and  so 
on  to  the  end. 

EXAMPLES  XXIII. 
Written  Exercises. 

Find  the  L.C.M.  of 

1.  357,391.  3.   3497,4035.  5.   165,198,242. 

2.  851,943.  4.   4899,5893.  6.   312,429,572. 

7.  360,  1350,  1500.         9.   420,  630,  1050,  and  1470. 

8.  195,  546,  286.  10.    1365,  2288,  2.640. 


Arts.  97-101.]  PARENTHESIS.  79 

98.   By  using  the  form  of  work  as  given  in  Art.  95,  we 
may  find  the  H.C.F.  and  the  L.C.M.  at  the  same  time. 

Ex.     Find  the  H.C.F.  and  the  L.C.M.  of  24,  60,  and  72. 

24  =  23  x  3 

60  =  22  x  3  x  5  ' 

72  =  23  x  32 


H.C.F.  =  22  x  3  =12. 

L.C.M.  =  23x  32  x  5  =  360. 

It  seems  best  to  make  a  digression  here  to  explain  the 
use  of  the  sign  of  parenthesis,  and  the  use  of  cancella- 
tion. 

Parenthesis. 

99.   The   sign  of  parenthesis  is  made  thus,  (  ) ;   in 
mathematics  this  sign  is  often  called  parenthesis. 

100.  Several  numbers  are  included  in  parenthesis  when 
they  are  to  be  treated  as  a  tvhole,  some  sign  being  put  outside 
the  parenthesis  to  show  hoiv  this  whole  is  to  be  treated. 

Thus,  17  +  (4+8)  denotes  that  8  is  to  be  added  to  4  and  that 
this  result  is  then  to  be  added  to  17. 

Again,  117  —  6  x  (4  +  8)  denotes  that  6  is  to  be  multiplied  by  the 
sum  of  4  and  8,  and  that  the  product  is  to  be  subtracted  from  117. 
Also,  36  -=-  (24  -f-  4)  denotes  that  24  is  to  be  divided  by  4,  and  that 
36  is  to  be  divided  by  this  result. 

When  two  parentheses  come  together  without  any  sign 
between  them,  the  sign  of  multiplication  must  be  under- 
stood. 

Thus,  (5  +  7)  (9  -  3)  is  put  for  (5*  +  7)  x  (9  -  3). 
Also,  6  (5  +  7)  is  put  for  6  x  (5  +  7). 

101.  Instead  of  enclosing  numbers  in  parenthesis  a 
line,  called  a  vinculum,  is  sometimes  drawn  over  them. 

Thus,  17  +  4+8  may  be  used  instead  of  17  +  (4  +  8). 


80  FACTORS  AND   MULTIPLES.  [Chap.  III. 

102.  Sometimes  a  parenthesis  is  put  within  a  paren- 
thesis :  to  avoid  confusion  the  parentheses  are  made  of 
different  shapes  and  are  named  as  follows : 

(  ),   sign  of  parenthesis, 
j  j,   braces, 
[  ],   brackets. 

For  example,  150  -  3  {13  -  (9  -  3)}. 

In  order  to  simplify  when  there  is  more  than  one  bracket,  it  will 
be  found  convenient  to  clear  away  the  innermost  bracket  first. 
Thus,  150  -  3  {13  -  (9  -  3)}  =150-3  {13  -  6}  =  150  -  3  x  7 
=  150  -  21  =  129. 

Operations  of  multiplication  and  division  are  to  be 
performed  in  order  from  left  to  right,  and  each  sign  is  a 
direction  to  multiply  or  divide  by  the  number  that  follows 
next  after  it. 

For  example,       60  +    6x3=    10  x  3  =    30  ; 
and  60  x    6  -  3  =  360  -  3  =  120  ; 

and  60-10-2=      6-2=      3. 

EXAMPLES  XXIV. 


1.  8+  (7+3). 

2.  15  -(9  +  3). 

3.  27 -(11 -4). 

4.  15-2(8-5). 

5.  3  7-2-4(8- 

6.  7(3 +  9) -5  (12 

13.  325-(17-2)(24-5). 

14.  27  -  3(20  -  11)  +  4(8  -  3). 

15.  18+S17-2-(15-4)J. 

16.  23 -[41-  {2  +  lj  -27^6]. 


Written  Exercises. 

7. 

(.9  +  .7)(.5  +  l.l). 

8. 

(13- 

.5)(1.5- 

-.7). 

9. 

(23- 

12)  (28  - 

-12). 

10. 

3(13- 

-4)(15- 

-7). 

6). 

11. 

8(1.7 

-1.1)(1. 

3  -  .06), 

-4).- 

12. 

5(11  +  5)(11- 

-5). 

Arts.  102-104.]  CANCELLATION.  81 

17.  (3.8016  -2.794) (1.8093 -. 078). 

18.  12-6  x  2. 

19.  12  +  (6x2).  22.    28+- 7-3  +  2.  • 

20.  18x6 +-2.  23.    28 +-(7 -3) +2. 

21.  18x(6+-2).  24.    28  +-  [7  -  (3  +  2)]. 

Cancellation. 

103.  Here  we  make  use  of  the  following  principle  : 

Dividing  both  dividend  and  divisor  by  the  same  number 
does  not  change  the  quotient. 

Thus,  24  -7-  4  =  6,  and  if  both  24  and  4  be  divided  by  2,  we  shall 
have  12  -4-  2  =  6  ;  also,  if  both  24  and  4  be  divided  by  10,  we  shall 
have  2.4  -f-  .4  =  6. 

104.  Note.  It  should  be  noticed  that  although  the  quotient 
is  unchanged  by  dividing  both  dividend  and  divisor  by  the  same 
number,  the  remainder,  if  any,  is  not  unchanged,  but  is  equal  to 
the  original  remainder  divided  by  the  number  by  which  the  original 
divisor  and  dividend  were  multiplied. 

Therefore  we  must  multiply  the  remainder,  if  any,  by  the  number 
used  in  dividing,  if  we  wish  the  remainder  obtained  by  using  the 
original  numbers. 

Ex.  1.  44  -4-  8  =  5,  with  a  remainder  of 4  ; 

while  ll--2  =  5,    "     "  "  "  1. 

It  is  evident  that  the  remainder  1  compared  with  the  divisor  2  is 
just  as  large  as  the  remainder  4  compared  with  the  divisor  8. 

Ex.  2.  How  many  pieces,  each  24  inches  long,  can  be  cut  from  a 
string  231  inches  long,  and  what  will  be  the  length  of  the  part  left 
over  ? 

231  -*-  24  =  77  +  8  =  9,  with  a  remainder  of  5. 

This  remainder  must  be  multiplied  by  3,  if  we  wish  to  know  how 
many  inches  of  string  remain.  Ans.  =  9  pieces,  with  15  inches 
remaining. 


82  FACTORS  AND   MULTIPLES.  [Chap.  III. 

105.  The  principle  of  Art.  103  may  be  used  in  shortening  the 
process  of  division,  especially  when  the  dividend  and  divisor  can  be 
factored  at  sight.    Thus, 

Ex.  1.     99)3663  gives  the  same  quotient  as  11)407 

37  =  Ans. 
Ex.  2.    36)4884  =  6)814 

135,  rem.    4. 
4884  -f-  36  =  135,  rem.  24.  [Compare  Art.  68.] 

106.  Division  has  thus  far  been  indicated  by  the  sign 
-h  ;  but  division  is  often  indicated  by  writing  the  dividend 
just  above  the  divisor  with  a  line  between ;  thus, 

36  -7-  8  is  sometimes  written  ^-, 

and  what  is  true  of  the  first  expression  is  true  of  the 

second.     In  either  case  dividing  both  36  and  8  by  the 

same  number  does  not  change  the  quotient ;  thus, 

36-j-8  =  9-j-2  =  4,  with  a  remainder  of  1.  )   .    ,   _,  A . 

Art.  104. 


¥  = 


=  f         =4, 


of  1. ") 
"  I-  > 


107.   The  process  of  dividing  both  dividend  and  divisor 
(or  their  factors)  by  the  same  factor  is  called  Cancellation. 
A  thin  line  drawn  through  a  dividend  or  divisor  indicates  that  a 
factor  has  been  cancelled  ;  thus, 
20 
^  =  —  =  2,  with  a  remainder  of  6. 

n     7 

7 
Here  6  is  cancelled  from  dividend  and  divisor,  and  the  quotients 
are  written,  one  above  and  one  below. 
Ex.    (5  x  48  x  28)  +  (10  x  14  x  9)  may  be  written 
5  x  48  x  28 
10  x  14  x  9* 
The  latter  is  the  better  form  when  we  wish  to  cancel ;  thus, 

1      16        i 

$  x  S.  x  8  - 1?  =  5,  with  a  remainder  of  1. 

Xfixlix  9     3 

%        %        3 


Arts.  105-107.]    MISCELLANEOUS  EXAMPLES.  83 

Here  5  is  cancelled  from  5  and  10,  3  from  48  and  9,  7  from  28 
and  14,  and  the  two  2's  thus  obtained  in  the  divisor  cancel  with 
the  4  in  the  dividend.  The  result  is  the  same  as  if  there  had  been 
no  cancellation. 

EXAMPLES    XXV. 


Written  Exercises. 

\m 

lplify 

by  cancellation : 

l. 

» 

4-    *fc 

7-    t7A- 

10. 

444 
T6T' 

2. 

» 

5-    iff 

8.  m- 

11. 

tu- 

3. 

if- 
13. 

6.  iH- 
7x22 
11  x  63" 

9-    Iff 

12. 

rn- 

14. 

(8  x  38  x  41)  - 

(19  x  4). 

15. 
16. 

6(42  -  3) 

13x8  ' 

1.26  x  3.5 

.6x7 
17.   Why  can  we  not  cancel  in  "  . 


EXAMPLES    XXVI. 
Miscellaneous  Examples,  Chap.  IU. 

1.  Divide  the  product  of  8978  and  55112  by  5561. 

2.  Divide  210  dollars  between  A  and  B,  so  that  A 
may  have  5  times  as  much  as  B. 

3.  Find  the  H.C.F.  of  3465  and  3696. 

4.  Multiply  606.78  by  11. 

5.  Find352;  1052;  7.52;  V24™09. 

6.  What  number  is  the  same  multiple  of  7  that  21560 
is  of  55? 


84  MISCELLANEOUS  EXAMPLES.        [Chap.  III. 

7.  What  is  the  price  of  a  silver  bowl  weighing  50 
ounces,  at  1.25  dollars  an  ounce  ? 

8.  Two  equal  sums  were  respectively  divided  among 
12  men  and  a  certain  number  of  boys.  Each  man  received 
5  dollars,  and  each  boy  1  dollar.  How  much  was  divided 
altogether,  and  how  many  boys  were  there  ? 

9.  Exactly  20  years  ago,  a  man  was  four  times  as  old 
as  his  son,  whose  present  age  is  28.  What  is  the  present 
age  of  the  father  ? 

10.    Eindl9  x  16;  656x125. 


11.  A  certain  chapter  of  a  book  begins  at  the  top  of 
the  357th  page  and  ends  at  the  bottom  of  the  435th  page. 
How  many  pages  are  there  in  the  chapter  ? 

12.  After  multiplying  375  by  29,  and  131  by  some 
other  number,  the  results  when  added  amounted  to  13888. 
What  was  the  other  number  ? 

13.  Find  the  H.C.F.  of  5610,  11781,  and  1309. 

14.  Find  the  least  number  by  which  222  must  be 
multiplied  in  order  that  the  product  may  be  a  multiple 
of  1295. 

15.  Four  bells  toll  at  intervals  of  3, 4,  5,  and  6  seconds, 
respectively.  If  they  all  begin  to  toll  at  the  same  instant, 
how  long  will  it  be  before  they  again  all  toll  together  ? 


16.  Add  fourteen  hundred  seventeen,  four  thousand 
eleven  hundred  nine,  six  million  fifteen  thousand,  and 
eighteen  million  twelve  hundred  nineteen. 

17.  A  certain  number  was  divided  by  35  by  '  short ' 
divisions ;  the  quotient  was  72,  the  first  remainder  was 
2,  and  the  second  remainder  was  6.  What  was  the 
dividend  ? 


Art.  107.]         MISCELLANEOUS    EXAMPLES.  85 

18.  Multiply  700630.0003  by  1006.07,  and  prove  by 
dividing  the  product  by  the  multiplier. 

19.  Find  the  continued  product  of  18,  13,  and  11 ; 
obtain  the  square  root  of  the  product  to  two  decimal 
places. 

20.  Divide  126819  by  21,  using  factors. 


21.  What  is  the  least  number  of  times  that  315  must 
be  added  to  1594  that  the  sum  may  exceed  a  million  ? 

22.  Multiply  67412  hf  9997  as  shortly  as  you  can. 

23.  Divide  789  by  .10063  to  3  decimal  places. 

24.  Find  the  H.C.F.  of  10481  and  17617. 

25.  Four  men  can  walk  30,  35,  40,  and  45  miles  a  day, 
respectively ;  what  is  the  least  distance  they  can  all  walk 
in  an  exact  number  of  days  ? 


26.  Find  the  L.C.M.  of  12,  64,  80,  96,  120,  160. 

27.  Find  the  prime  factors  of  1176  and  19404,  and 
hence  write  down  their  G.C.M.  and  L.C.M. 

28.  The  quotient  is  twice  the  divisor,  and  the  remainder 
which  is  50  is  one-fifth  part  of  the  quotient.  Find  the 
dividend.    . 

29.  Simplify   ;   obtain  the  answer   in  two 

forms.  125X219' 

30.  Find  the  least  number  which  can  be  divided  by 
7,  20,  28,  and  35,  and  leave  3  as  remainder  in  each  case. 


31.  What  number  is  that  which  after  being  subtracted 
19  times  from  1000  leaves  a  remainder  of  12  ? 

32.  Multiply  three  thousand  eighty-seven  by  seventy- 
two  thousand  nine  hundred  thirty.  What  numbers  less 
than  12  will  exactly  divide  the  product  ? 


86  MISCELLANEOUS  EXAMPLES.     [Chaps.  III.,  IV. 

33.  (a)    Simplify  650  x  1.25  -  .5. 

(b)   The   answer  is  a  multiple  of  which  of  the 
following  numbers  :  5,  15,  25,  65,  105,  125  ? 
Obtain  (6)  by  first  obtaining  primes  of  the  answer. 

34.  Find  19  x  17  x  11  X  2.5  x  1.25. 

35.  Find652x  .11. 

36.  Simplify  (a)  \2-  x  4  -  2  +  6  (18  -  14). 

(P)  ¥-X  (4-2) +  6(18-14). 
(c)  2(¥x4)  -.J2  +  6(18  -14)j. 


37.  If  a  number  when  divided  by  35  give  a  remainder 
27,  what  remainder  will  it  give  when  divided  by  7  ? 

38.  What  is  the  greatest  and  what  is  the  least  number 
of  four  digits  which  is  exactly  divisible  by  73  ? 

39.  Find  the  H.C.F.  and  the  L.C.M.  of  21,  22,  24,  28, 
32,  33;  also  of  16,  18,  20,  24,  30,  36. 

40.  Find  the  number  nearest  to  1000   and   exactly 
divisible  by  39. 

41.  Multiply  7863  by  999,  and  see  if  the  product  is 
divisible  by  3. 

42.  Find  V4912.888464. 

43.  Find  V^6- 

(a)  Divide  the  following  numbers  by  2. 

(b)  Prove    your    answers    by    first    simplifying   the 
numbers,  and  then  dividing  by  2. 

44.  3(6  +  8);  3(6x8). 

45.  4(6  +  8);  4(6x8). 

46.  4(18  +  6);  4(18-6). 

47.  (6x2)  (8  +  10). 

48.  6(12-3) +  8  6  +  4  +  2. 

49.  28  -f-  [7  -  (3  +  2)]. 


Arts.  108-110.]  FRACTIONS.  87 


CHAPTER  IV. 

FRACTIONS. 

108.  If  a  unit  be  divided  into  2,  3,  4,  5,  etc.,  equal  parts, 
these  parts  are  called  halves,  third-parts,  fourth-parts, 
fifth-parts,  etc.,  or  more  shortly  and  more  generally,  halves, 
thirds,  fourths,  fifths,  etc. 

If  the  unit  quantity  be  divided  into  any  number  of 
equal  parts,  one  or  more  of  these  parts  is  called  a  Fraction 
of  the  unit. 

For  example,  if  a  unit  quantity,  as  one  apple,  be  divided  into 
sevenths,  three  of  these  parts  constitute  three  sevenths,  and  the 
three  sevenths  is  a  fraction  of  seven  sevenths,  the  unit  quantity. 

109.  The  number  which  indicates  how  many  parts  of 
the  unit  quantity  are  to  be  used  is  called  the  Numerator. 

The  number  which  indicates  into  how  many  parts  the 
unit  quantity  is  divided  is  called  the  Denominator. 

110.  The  expression  formed  by  writing  a  numerator 
just  above  a  denominator  with  a  line  between  is  called  a 
Common  Fraction. 

Thus,  f,  y8^  (eight-thirteenths),  ^  (one  twenty-third),  are  com- 
mon fractions  (called  briefly  fractions). 

Common  fractions  are  also  called  vulgar  fractions. 

Note.  A  fraction  is  an  expression  of  division,  the  numerator 
and  denominator  corresponding  to  the  dividend  and  divisor  respec- 
tively. What  is  true  of  dividend  and  divisor  is  true  of  numerator 
and  denominator.  When  the  indicated  division  is  performed,  the 
quotient  is  generally  a  decimal. 


88  FRACTIONS.  [Chap.  IV. 

Ex.   &  =  3-s-24. 

24)3.000(.125 
24 

60 
48 
120 
120 

111.  If  we  have  3  units,  and  divide  each  of  them  into 
5  equal  parts,  and  then  take  one  of  the  parts  from  each 
divided  unit,  we  shall  take  one  part  out  of  every  five,  that 
is,  one-fifth  of  the  whole  three  units  j  but  each  of  the  parts 
is  one-fifth  of  a  single  unit  and  we  take  3  of  them :  we 
therefore  take  3  fifths  of  one  unit. 

Thus,  3  fifths  of  1  unit  is  the  same  as  1  fifth  of  3  units. 
Hence,  f,  which  by  definition  denotes  3  fifths  of  1  unit, 
may  also  be  considered  to  stand  for  1  fifth  of  3  units. 
The  same  holds  good  for  all  other  fractions  ;  for  example, 

f  of  1  dollar  =  £  of  3  dollars ; 
and  |  of  1  foot     =  }  of  7  feet. 

EXAMPLES  XXVII. 

1.  Write  in  figures  the  following  fractions:  five- 
ninths,  six-elevenths,  eleven  twenty-thirds,  sixteen  twenty- 
sevenths,  seventeen  ninety-firsts,  ninety-five  one  hundred 
fourths. 

2.  Write  in  words :   f,  f,  ^  A>  A>  Vb  &>  tVd  J*& 

112.  The  numerator  and  denominator  of  a  fraction  are 
called  its  Terms. 

When  the  numerator  is  less  than  the  denominator,  the 
fraction  is  called  a  Proper  Fraction ;  and  when  the  numera- 
tor is  equal  to  or  greater  than  the  denominator,  the  fraction 
is  called  an  Improper  Fraction. 


Arts.  111-114.]  MIXED   NUMBERS.  89 

A  number  made  up  of  an  integer  and  a  fraction  is  called 
a  Mixed  Number. 

Thus,  2\  (2  and  \),  which  means  2  +  £,  is  a  mixed  number. 

Changing  the  form  of  an  expression,  or  changing  the 
units  in  terms  of  which  any  quantity  is  expressed,  is 
called  Reduction. 

113.  Reducing  a  mixed  number  to  an  improper  fraction. 

Consider,  for  example,  3|. 

Each  unit  contains  7  sevenths,  therefore  3  units  contain  3  times 
7  sevenths. 

21-4-2      23 
Hence,  31  =  3  times  7  sevenths  +  2  sevenths  =  — ±—  =  — 

7  7  7 

a™^                           72      7x9  +  2      65 
Again,  7f  = =  -• 

From  the  above  it  will  be  seen  that  a  mixed  number  is  equivalent 
to  an  improper  fraction  whose  denominator  is  the  denominator  of 
the  fractional  part,  and  whose  numerator  is  obtained  by  multiplying 
the  integral  part  by  the  denominator  of  the  fraction  and  adding  its 
numerator. 

It  should  be  noticed  that  a  whole  number  can  be  expressed  as 
a  fraction  with  any  given  denominator.     For  example, 

6  =  6x7  sevenths  =  -472- ;  also,  6  =  6  x  13  thirteenths  =  f  f . 

114.  Conversely,  reducing  an  improper  fraction  to  a 
whole  or  mixed  number. 

Consider,  for  example,  -2^. 

Since  7  sevenths  make  1  unit, 

-2f  =  3x?  sevenths  +  2  sevenths  =  3  +  2  sevenths  =  3f . 

Again,  -2¥*  =  6x4  fourths  =  6,  since  4  fourths  =  1. 

From  the  above  it  will  be  seen  that  an  improper  fraction  is 
reduced  to  a  mixed  number  by  dividing  its  numerator  by  its  denom- 
inator ;  the  quotient  will  form  the  integral  part,  while  the  remainder, 
if  any,  will  form  the  numerator  of  the  fractional  part,  whose 
denominator  must  be  the  denominator  of  the  improper  fraction. 


90  FRACTIONS.  [Chap.  IV. 

EXAMPLES  XXVIII. 
Oral  Exercises. 

Express  as  improper  fractions : 
1.   11  If,  2|,  3*.  2.   7^,  6T%  5f  3f 

3.  4^,  9f,  12fc  11^. 

4.  Express  3, 5,  and  9  as  fractions  with  a  denominator  7. 

Express  as  whole  or  mixed  numbers : 

s.  ¥> ¥>¥>¥•   e.  ft  ?>  ft  if   7.  ¥>¥>¥>«• 

Written  Exercises. 

Express  as  improper  fractions : 
8.  4^,5^,18^.     [Art.  55,1.] 

10.  Reduce  13  to  fifteenths,  and  41  to  twenty-fifths. 

11.  Express  427  as  a  fraction  with  a  denominator  99. 

Express  as  whole  or  mixed  numbers : 

12-    ft  ft  If      13.    tflifoW      14-    ftW,W- 
See  Art.  70  for  examples  in  14. 

115.   Reducing  a  fraction  to  its  lowest  terms. 
A  fraction  is  said  to  be  in  its  Lowest  Terms  when  the 
numerator  and  denominator  have  no  common  factor. 

Thus,  the  fractions,  §,  f ,  |§,  are  in  their  lowest  terms ;  but  the 
fractions,  f ,  |£,  § £ ,  are  not  in  their  lowest  terms,  for  in  each  case 
the  numerator  and  denominator  have  2  as  common  factor. 

The  following  is  a  very  important  truth : 

The  value  of  a  fraction  is  not  changed  by  dividing  the 
numerator  and  denominator  by  the  same  number. 

This  truth  is  but  a  repetition  of  the  principle  stated  in  Art.  103. 


Art.  115.]        REDUCTION  TO  LOWEST  TERMS.  91 

Ex.   B educe  T%\5u  to  its  lowest  terms. 

To  reduce  to  the  lowest  terms  we  must  divide  by  the  H.C.F.  of 
the  numerator  and  denominator ;  for  we  thus  obtain  an  equivalent 
fraction  whose  numerator  and  denominator  have  no  common  factors. 

In  the  present  case  the  H.C.F.  will  be  found  to  be  55. 
825       825  -55      15 


Thus, 


1540     1540  ■*■  55     28 


Instead  of  reducing  a  fraction  to  its  lowest  terms  by  dividing 
the  numerator  and  denominator  by  their  H.C.F.,  we  may  divide 
by  any  common  factor,  and  repeat  the  process  until  the  fraction  is 
reduced  to  its  lowest  terms.    Thus, 

We  see  at  once  that  5  is  a  common  factor ;  we  therefore  divide 
the  numerator  and  denominator  by  5,  and  obtain  the  equivalent 
fraction  iff.  We  now  see  that  11  is  a  common  factor,  and  having 
divided  the  numerator  and  denominator  by  11,  we  have  the  equiva- 
lent fraction  £f ,  which  is  at  once  seen  to  be  in  its  lowest  terms. 


EXAMPLES    XXIX. 
Oral  Exercises. 

Reduce  to  their  lowest  terms : 

*•     ¥>  "6"?  T5"?  T6">  T5-  3*     "2T>  "5T>  JTi  H' 

2-  «,«,*,  if,  it  4.  ft  ft  ft  ft,  ft 

Written  Exercises. 

Reduce  to  their  lowest  terms : 

»•  fit      s-  m      ii-  mi-    i4.  tut 
e-  m-      9-  m      i2-  \m-    is-  Hft 

»•  m       io.  AVt        is-  A%-      is-  A-2A- 


92  FRACTIONS.  [Chap.  IV. 

116.  Reducing  fractions  to  equivalent  fractions  having 
the  lowest  common  denominator. 

The  following  is  a  very  important  truth : 

The  value  of  a  fraction  is  not  changed  by  multiplying  the 
numerator  and  denominator  by  the  same  number. 

This  truth  is  but  a  repetition  of  the  principle  stated  in  Art.  67. 

Consider  the  fractions,  f ,  f ,  and  f . 

The  L.C.M.  of  the  denominators  4,  6,  and  9  is  easily  seen  to 
be  36.  Since  36  is  a  multiple  of  each  denominator,  all  the  fractions 
can  be  reduced  to  equivalent  fractions  which  have  36  for  denom- 
inator, provided  the  numerator  and  denominator  of  each  of  the 
fractions  be  multiplied  by  a  suitable  number,  namely,  by  the  num- 
bers 36  -=-  4,  36  -r-  6,  and  36  -4-  9,  respectively  ;  that  is,  by  9,  6,  and 
4,  respectively. 


Thus, 


and 


3 

3 

x 

9 

27 

4 

4 

X 

9 

36 

5 

5 

X 

r> 

30 

6 

6 

X 

6 

36 

8 

8 

X 

4 

32 

9 

9 

X 

4 

36 

Again,  reduce  £},  ^r,  ^?,  to  equivalent  fractions  having  the  lowest 
common  denominator. 


Full  Work  Illustrated. 


18  =  2   x32 
30  =  2    x  3   x  5 
24  =  23  x  3 


L.C.M  =23x  32  x  5 


11  11  x  2*  x  5 

220 

18  18  x  22  x  5 

360* 

7   7  x  22  x  3 

84 

30  30  x  22  x  3 

360' 

5   5x3x6 
24  24  x  3  x  5 

_  75 
360 

117.   Comparison  of  Fractions. 

Of  two  fractions  which  have  the  same  denominator,  the  greater 
is  that  which  has  the  greater  numerator ;  for,  the  parts  being  the 
same,  the  greater  fraction  is  that  which  has  the  most  of  them. 


Arts.  116,  117.]  EXAMPLES.  93 

Again,  of  two  fractions  which  have  the  same  numerator,  the 
greater  is  that  which  has  the  smaller  denominator  ;  for,  the  number 
of  parts  being  the  same  in  both,  the  greater  is  that  in  which  the 
parts  are  the  greater  ;  that  is,  in  which  the  unit  has  been  divided 
into  the  smaller  number  of  equal  parts. 

We  can  therefore  see  at  once  which  of  a  number  of  fractions  is 
the  greatest,  and  which  is  the  least,  provided  the  fractions  are  first 
of  all  reduced  to  equivalent  fractions  with  the  same  denominator. 
For  this  particular  purpose  it  would  do  equally  well  to  reduce  the 
fractions  to  equivalent  fractions  with  the  same  numerator,  but  it 
is  for  other  purposes  much  less  convenient  to  reduce  fractions  to 
equivalent  fractions  with  the  same  numerator. 

Ex.    Which  is  the  greatest  and  which  is  the  least  of  the  fractions, 

As  in  the  preceding  article,  the  fractions  are  equivalent  to  f|,  § £ , 
and  ||,  respectively;  they  are  therefore  in  ascending  order  of 
magnitude. 

EXAMPLES    XXX. 

Written  Exercises. 

Eeduce  to  equivalent  fractions  with  the  lowest  common 
denominator,  and  arrange  in  ascending  order  of  magni- 
tude: 

*•    h  |>  T  5-    t>  A>  A* 

*•    "3~>  9?  TTm  6.    y^,  -Jy,  -g-g-. 

3-   h  A?  A-  7-    J?  "fi  p 

!3-    Afiff  16. 

I4-    hh  iA>*<  17. 

is-  A>  «>**>»  is-  f>A>«,H,ffr 

Eeduce  to  equivalent  fractions  which  have  the  lowest 
common  numerator : 

19        5.10.15  Oft        121618  oi         15     18      20 

xy'     8'   IT?   2T'  *"•     TT>  T9>  ft'  /J1'     ST>  ft?  if- 


9. 

2     3      5 
3"?  T>  T 

10. 

i,  h  A- 

11. 

H>  if.  H- 

12. 

**»  it,  It- 

4 

A, 

A.  it- 

3       5         7       11 
8?  TT>  TS>  TO"' 

94 


FRACTIONS. 


[Chap.  IV. 


118.   Addition  of  Fractions. 

Fractions  which  have  the  same  denominator  are  called 
Similar  Fractions. 

If  fractions  are  dissimilar  they  must  be  made  similar 
[Art.  116] ;  then  their  numerators  may  be  added,  and  the 
sum  written  as  a  numerator  for  the  common  denominator. 
[Compare  Art.  32.] 

Ex.  1.    Find  f  +  TV  +  f  • 


5  5x6       30 

6  6x6       36 

Or, 

5_ 
6 

30 

7        7x3      21 
12      12  x  3     36 

7  _ 
12 

21 

4  _  4  x  4    _  16 
9      9x4       36 

4_ 
9 

16 

Sum=^  = 
36 

If*. 

Sum  = 

~67~ 
36 

m 

After  a  little  practice  the  middle  column  might  be  omitted. 
Ex.  2.    Find  24  +  3A. 

3&  =  »& 

Sum  =  6& 

Here  the  12ths  are  added,  and  1  is  carried  to  units.  The  pro- 
cess is  similar  to  that  represented  in  Ex.  2,  Art.  29. 

The  result  should  in  all  cases  be  reduced  to  its  lowest 
terms,  and  an  improper  fraction  should  be  expressed  as  a 
mixed  number. 

EXAMPLES   XXXI. 
Oral  Exercises. 

Find  the  sum  of  the  following  fractions : 

1.  £andf.  4.    JLand^.  7. 

2.  |  and  \.  5.   f  and  f.  8. 

3.  ^andT%.  6.    Handff-  9- 


i  and  f . 
|  and  £ 
f  and  f 


Art.  118.]  EXAMPLES.  95 

10.  f  andf  14.  20^andlOTV 

11.  f  andf  15.  3£and4£. 

12.  2Jand3f.  16.  8f  and  6  A 

13.  41^and61^.  17.  12^- and  6^. 

18.   4f  and6A- 

Written  Exercises. 

Find  the  sum  of 

19.  A  and  fa  22.    2|  and  1-^.  25.   2f  and  3|. 

20.  A  and  fa  23.   5|  and  2^.  26.    3f  and  1£. 

21.  AandA-  24.    7f  and  f  27.    Iff  and  7£f. 

28.  *,  ♦,  A,  and  «.  30.    A>  A>  A>  A>  tt>  **&  H ■ 

29.  f,f,f,andf.  31.    A  A>  "B>  tt  «>  and  ff . 

32.  Findf  +  A  +  A  +  A  +  A- 

33.  Find  #  +  A  +  *  +  &  +  & 

34.  Find3f  +  A  +  5A  +  7TV 

35.  Find  3J  +  A  +  5^  +  3^  +  7ff 

36.  Find  3J  +  5f  +  tt  +  3&. 

37.  Fmdff  +  2A  +  UA  +  5ff 

38.  Find3f  +  7A  +  A  +  2if. 

39.  Find  fttfr  +  BJfa 

40.  Find3AV  +  5AV  +  llMt- 

41.  Find  10jfj  +  ll^  +  7fff 


96  FRACTIONS.  [Chap.  IV. 

119.    Subtraction    of    Fractions. 

Subtraction  can  be  performed  with  fractions  only  when 
they  are  similiar.     [Compare  Art.  39.] 

Ex.  1.    Subtract  f  from  ft.  Ex.  2.    Find  6ft  -  3f. 

Difference  =  fT.  Remainder  =  2^T. 

Ex.  8.    Subtract  5\\from  5f. 

Here  f|  cannot  be  subtracted  from  £f, 

5f  =  Hi        therefore  we  take  1  unit  from  the  5  units  and 

3H  =  3|f        add  it  (changed  to  24ths)  to  ft,  making  §£ ; 

Remainder  =  lft.       now  II  from   f I  equals    ft,   and    3    from 

4  equals  1. 

The  operation  is  similar  to  that  represented  in  Ex.  2,  Art.  38. 

Ex.  4.    Simplify  3 J 


-  2f  +  8f  - 

-S&.-SA  +  &     [See  Art.  41.] 

3*=    % 

8|=    8ff 

2f=    2H 
*&  =    2ft 

12H 

-      io|f  =  iff. 

EXAMPLES   XXXII. 
Oral  Exercises. 

Simplify  (give  lowest  terms  in  your  answers) 


"*•  A— A- 

5-    f-f 

9. 

5     _  2 

2-    f-f 

6.    f-f. 

10. 

3A-V- 

3-  A- A 

7-    l-f 

11. 

6^-4|. 

4#      3"2   ~~  A* 

8-  A-f 

12. 

6*-2A 

Simplify ; 

Written  Exercises. 

13.  H-A 

15.    ft-» 

17. 

*-A 

"•  A- A 

is.  A-A- 

18. 

T2~  —  7* 

Arts.  119,  120.]  MULTIPLICATION.  97 

19.  A -A-        23-  3i~2i-        27-  7*-5A- 

20.    A  -A  24-  m-5U-         28.    19JT-12A- 

21-    A~»  25.  2f|-2||.         29.    r^-flA- 

22.    A~to  26.  3f-2Jy  30.    3^-2^. 

31.   6J-2||.  33.  16ft -5^ 

32-    5ft- Sft  34.  9ft-4ftU. 

Find  the  difference  between 

35.  3^- and  5^.  39.  8JJ-  and  12 Jf. 

36.  728Tand827¥.  40.  6fJ-  and  12JJ. 

37.  6^andl5|f  41.  143^  and  127^. 

38.  7iiand5||.  42.  85^r  and  72/^. 


Simplify : 

43. 

2i  +  3i-4i. 

44. 

GH-Si  +  IA- 

45. 

5|-3A  +  ^-2^. 

46. 

15^-13A  +  16A-9M- 

47. 

12^-10  +  7^-t-5f£. 

48. 

4f-2A  +  2H-3^. 

49. 

6A-|"2A  +  6rlr  +  TTfW 

120.   Multiplication  by  a  Whole  Number. 

Fractions  may  be  treated  as  concrete  numbers  ;  therefore, 
as  3  times  5  tons         =  15  tons, 

so  3     "     5  sevenths  =  15  sevenths  ; 

i.e.,  3     " .:   f  =¥• 

Again,       3      "     ■&  =  |f        =  f  (by  cancellation) 

5  5 


i.e.,  3 


18-5-3     6 


Hence,  to  multiply  a  fraction  by  a  whole  number,  we  must 
multiply  the  numerator,  or  (when  possible)  divide  the  denomina- 
tor, by  that  whole  number. 


98  FRACTIONS.  [Chap.  IV. 

The  product  should  always  be  reduced  to  its  lowest  terms,  and 
an  improper  fraction  should  be  expressed  as  a  mixed  number. 

Ex.     Multiply  T5g  by  15. 

±xid=*kH=I5  ==?*=*  4*. 

18  18  18       6         * 

EXAMPLES   XXXIII. 
Oral  Exercises. 

Multiply  and  reduce  to  their  simplest  forms : 

1.  *  by  2.  3.    24iby3.  5.   f  by  4. 

2.  ^- by  3.  4.    T3Tby4  6.   &  by  4. 

7.  |  by  3.  9.    ^by6. 

8.  |f  by  17.  10.   ^by8. 

Written  Exercises. 

Perform  the  following  examples  (see  Art.  107) : 

11.  tVxlO.        15.   7T\x  25.  19.   -V/X22. 

12.  T%  x  8.  16.   |f  x  15.  20.   44^  x  26. 

13.  2|x6.  17.    9fix25.  21.   99  x  ^. 

14.  5f  x  10.        18.    ||  x  16.  [Art.  55.]   22.    ^ffg  x  9. 

121.   Multiplication   by   a   Fraction. 

We  understand  multiplication  to  be  the  taking  one 
number  as  many  times  as  there  are  units  in  another. 
Thus,  to  multiply  5  by  4,  we  take  as  many  fives  as  there 
are  units  in  4.  Now  4  is  1  +  1  +  1+1,  and  5  x  4  is 
5  +  5  +  5  +  5. 

Thus,  to  multiply  one  number  by  a  second  is  to  do  to  the 
first  what  is  done  to  the  unit  to  obtain  the  second. 

For  example,  to  multiply  f  by  f ,  we  must  do  to  f  what  is  done 
to  the  unit  to  obtain  | ;  that  is,  we  must  divide  f  into  4  equal  parts 


Art.  121.]  MULTIPLICATION.  99 

and  take  3  of  those  parts.     Each  of  the  4  parts  into  which  $  is 

divided  will  he  — — ,  and  hy  taking  3  of  these  parts  we  get     x    • 
7x4  7x4 

Thus,  ^x3      5x8 


7      4      7x4 

Hence,  the  product  of  any  two  fractions  is  another  fraction 
whose  numerator  is  the  product  of  their  numerators  and  whose 
denominator  is  the  product  of  their  denominators. 

The  continued  product  of  any  number  of  fractions  is 
obtained  by  continued  application  of  the  above  rule. 

Thus,  to  find  the  continued  product  of  §,  f ,  and  |. 

2  x  i  x  -  =  2  x4  x  -  =  2  x4x8  =  i*£. 

3  5   9   3x5   9   3x5x9   135* 

Hence,  the  product  of  any  number  of  fractions  is  another  frac- 
tion whose  numerator  is  the  product  of  their  numerators  and 
whose  denominator  is  the  product  of  their  denominators. 

It  should  be  noticed  that  the  product  of  one  fraction  by 
a  second  is  equal  to  the  product  of  the  second  by  the  first. 

It  should  be  noticed  also  that  an  integer  x  a  fraction 
equals  (the  integer  x  the  numerator)  h-  the  denominator. 

Ex.  1.   Multiply  j%  by  fy. 

2      1 

ixl=M_=i.  [Art.  107.] 
35      27     3^x2/     45 
5       9 

Ex.  2.    Simplify  f  X  f  X  f. 

1      1      1 

3X^X5~5 

1      % 
1 

Ex.  3.   Multiply  2\  by  3|. 

The  mixed  numbers  must  first  be  reduced  to  improper  fractions. 

Thus,  2}x3!  =  ?x2-9  =  9A-?9  =  261 

*       *4      8       4x8       32 


100  FRACTIONS.  [Chap.  IV. 

EXAMPLES   XXXIV. 
Simplify:  °ral  Exercises. 

MX}-  7.  |X|.  13.  |xf{. 

2.  }xf  8.  fx*f  14.  ||x|f 

3.  fxf.  9.  fx||.  15.  «XA- 

4.  fxf  10.  fx&  !6-  J|XA- 

5.  Hx21  ii.  _Z_X22.  17.  (|)2. 

6.  2J  X  3|.  12.  f  X  ft.  18.  (f)2.    19.  (|)3. 

Written  Exercises. 

20.  5^X3^.  27.  frx5&x{i. 

21.  |  X  |  X  f  28.  2TV  X  3f  X  6£. 

22.  fxfxf  29.  1^X21X1^. 

23.  fxAxA-  30.  6TVx||xl|f. 

24.  «xl|x5f  31.  ^  x  2^  x  5Jr  x  TW 

25.  2^X3^X4^.  32.    1^  X  Iff  X  f$  X  l^V 

26.  5ix71x^.  33.  ttxl|x5^x2|f. 

34.    (|)«.  35.    (A)2.  36.    (#)«. 

122.   Division  by  a  Whole  Number. 

Just  as  15  tons  h-  3  =  5  ions, 

so  also  15  sevenths  -*■  3  =  5  sevenths, 

that  is,  -1/  h-  3  =  f 

Again,  to  divide  f  by  3. 

5  5x3 
Here  5  is  not  a  multiple  of  3.     But,  since  -  =  — - — , 

6  6x3 

5^  o  _  5x3.  g  _     5     _  5 
6  '      ~  6  x  3  '      ~  6  x  3  ~  18* 


Art.  122.] 


EXAMPLES. 


101 


We  see  at  once,  that   a  fraction  is  divided  by  a  whole  number 
by   multiplying  its   denominator    by    the    whole    number.     For 


example,  in 


x3 


there    are  the    same    number  of    parts  as  in 


|,  namely  five,  but  the  unit  in  the  former  case  is  divided  into  3 
times  as  many  parts  as  in  the  latter,  and  therefore  each  of  the 

parts  in is  one-third  of  each  of  the  parts  in  4. 

F  6x3  * 

Hence,  to  divide  a  fraction  by  a  whole  number,  we  must  divide 
the  numerator,  or  multiply  the  denominator  (only  when  neces- 
sary), by  that  whole  number. 


Ex.  1.    Divide  31  by  7. 


34-  -  7  =  ^  -r-  7 
5  5 


16 


5x7 


16 
35' 


Ex.  2.   Divide  215f  by  9. 


23  +  8f  -=-  9 
23 


When  the  integral  part  of  the  dividend 


=  23M. 


is  large,  we  first  divide  the  integer  by  the 
»  "*"      divisor  ;   then  the  remainder  +  the  frac- 
tional part  is  to  be  divided  by  the  divisor. 


Simplify : 

1.  f  +  2. 

2.  i+3. 

3.  A  +  4- 

4.  ^  +  3. 


EXAMPLES   XXXV. 
Oral  Exercises. 


5.  f-r-8. 

6.  ^^4. 

7.  M-s-5. 


ft        42  _l_  7 


9.   ^  +  25. 


10.   H 


11.  a  +  12. 

12.  H 


17.  [Art.  55.] 


Written  Exercises. 


13. 

tt+6. 

17. 

2**11 

14. 

11  +  16. 

18. 

71-6. 

15. 

If +  30. 

19. 

9|-s-8. 

16. 

M  +  7. 

20. 

«  +  !& 

21.  5f-r-5. 

22.  7I--6. 

23.  8T\  +  15. 

24.  12f-f.ll 


102  FRACTIONS.  [Chap.  IV. 

25.    85$-*- 9.  26.    214TV-r-7.        27.    174f£  - 18. 

28.    7|i-r-15.  29.    254^-25. 

123.   Division  by  a  Fraction. 

If  the  fraction  -J  be  divided  by  1  (unity),  the  quotient 
is  -J;  but,  if  the  unit  be  separated  into  thirds,  and  one 
of  these  thirds  be  used  (instead  of  unity)  as  the  divisor, 
the  quotient  is  3  times  as  large  as  before. 

Thus,    |  +  l=fj  but  |  +  *  =  J  x  3  =  JyL. 

Now,  if  the  second  divisor  (i)  be  multiplied  by  2,  the  quotient 
(y)  must  be  divided  by  2  ;  thus, 

|  4-  f =    f  x  3  -^  2 

The  same  reasoning  will  apply  to  all  cases. 

Hence,  to  divide  by  a  fraction,  we  must  multiply  by  the  fraction 
inverted. 

Note.  Sometimes  a  short  method  of  dividing  a  fraction  by  a 
fraction  is  to  divide  the  numerator  and  denominator  of  the  dividend 
by  the  numerator  and  denominator  of  the  divisor,  respectively ; 

thus,  |  +  f '=  f . 

Ex.  1.  Divide  §  by  |f. 

1      16 
6^15  =  ^><^  =  16=17 
6  '  32     0     W      9        * 

3       3 

Ex.  2.   Dmde  2\  by  ljf. 

The  mixed  numbers  must  first  be  expressed  as  improper  fractions. 

1       9 


Arts.  123-125.]      COMPOUND   FRACTIONS. 


103 


Simplify : 


EXAMPLES   XXXVI. 
Written  Exercises. 


1. 

*-*-* 

12. 

4  _i_  2 
6"    *    "3' 

23. 

IT'S"    :    "5T" 

2. 

f+i 

13. 

2J+2J. 

24. 

121  _:_  143 

tt¥  •  rfv 

3. 

2  _t_  3 

9     '     4"* 

14. 

6A  +  1A- 

25. 

9  6      -11 

4. 

t  +  f 

15. 

6* -If. 

26. 

4_4    _s_  129 

5. 

i  +  f 

16. 

*H  +  iA- 

27. 

Ifl     |    1    7 

6. 

A  +  tt 

17. 

6|-11. 

28. 

1    23    _^_  1    37 
-'-TIT    *    J^TT^* 

7. 

*+* 

18. 

6f-=-9. 

29. 

143  _:_  959 

8. 

T?6^tV 

19. 

2  2     •    TT 

30. 

K7     •    913 

9. 

i-i-t 

20. 

2H^7. 

31. 

11^-12,% 

10. 

i  +  * 

21. 

«+*• 

32. 

2W+«Ht- 

11. 

t  +  i 

22. 

tt-*tt 

124.  When  unity  is  divided  by  any  number,  the  quo- 
tient is  called  the  Reciprocal  of  the  number ;  thus, 

l  is  the  reciprocal  of  5 ;  f  is  the  reciprocal  of  f ;  5  is  the  re- 
ciprocal of  \. 

Any  number  x  its  reciprocal  =  1. 

125.  A  fraction  of  a  fraction  is  called  a  Compound 
Fraction. 

Thus,  §  of  f  is  a  compound  fraction. 

To  take  f  of  f ,  we  must  divide  f  into  3  equal  parts  and  take  2  of 
those  parts. 

Hence,  §  of  f  is  the  same  as  \  x  f . 

Ex.  1.     Multiply  f  of  2i  by  f  of  If. 

f  of  2*  =  f  x  ¥»  sod  f  of  If  =*  f  x  i  j 

3      11      S      T      11 
hence  the  required  product  =  £x  —  x  "  x  -  =  — 

-  M      M      12 


104  FRACTIONS.  [Chap.  IV. 

EXAMPLES  XXXVII. 
Written  Exercises. 

1.    State  the  reciprocals  of  12,  f,  if-,  f  and  f^-. 
Simplify : 

2.  |  Off  9.  f  of  ^  of  ^. 

3.  |  off.  10.  21  of  31  off 

4.  fof^.  11.  6*  Of  2ft  Of  If 

5.  lfof2f  12.  |  off  X^of2J. 

6.  3iof6i.  13.  $  of  2f  x  1 A  of  21 

7.  7fof2f  14.  lfofajxiixof. 

8.  3£  of  3£.  15.  If  of  3|  x  5J  of  7f . 

126.   A  fraction  whose  numerator,  or  denominator,  or 
both,  are  fractional  is  called  a  Complex  Fraction. 

3        2.  1     I      1 

Thus,  JL,  JL,  and  2  Z  I  are  complex  fractions. 

Complex  fractions  are  simplified  by  dividing  the  numer- 
ator (simplified)  by  the  denominator  (simplified). 

Ex.  1.    Simplify  ±- 

j 

|      3  "  7     3     5      15 

Ex.  2.   Simplify  ttk. 
l+i 


0      ?      3 
3 

Caution.  Dividing  by  the  sum  of  two  fractions  is  not  equivalent 
to  multiplying  by  the  sum  of  the  reciprocals  of  those  fractions. 


Art.  126.]  COMPLEX   FRACTIONS.  105 

In  solviDg  the  above  example  the  following  would  be  wrong  : 

|^|=(m)x(t  +  f)- 

A  complex  fraction  is  unchanged  in  value  by  multi- 
plying its  numerator  and  denominator  by  the  same 
number. 


5 

g      9  =  45 

7       H     28 

4 


For 

example, 

5 
7  _ 

1" 

-fx 

11 
11 

For 

f" 

►*« 

fx 

1=1 

and 

IxiiH 

4x 

11  = 

Ex.  1, 


Multiply  the  numerator  and  denominator  by  24,  the  L.C.M.  of 
3,  4,  6,  8.    Then  we  have 

(f-f)24_18-126_2_9 
(f -|)  24      21-20      1 

Ex.  2.    Simplify   — 


5  +  — A 


7 2- 

4  +  i 

*  =  *• 


5  + =^-      5  + 


2         5  +  ^f 


4  +  i 

First,  multiply  the  numerator  and  denominator  of  the  lowest 
q 
complex  fraction,  namely  — ■ — ,  by  2,  and  we  get  f .     Next,  multiply 
4  +  i  o 

the  numerator  and  denominator  of  the  fraction by  9,  and  we 

7-1  o 

get  if.     Then  multiply  the  numerator  and  denominator  of  — - — 

5  + if 
by  57,  and  we  get  i£i,  which  is  then  reduced  to  its  lowest  terms. 

A  fraction  of  this  type  is  called  a  Continued  Fraction. 


106  FRACTIONS.  [Chap.  IV. 


EXAMPLES   XXXVIII. 


Simplify : 


1.    _.  4.    Z5.  7.    H.  10 


I  lo  i  H 


Written  Exercises. 

1  2 
10' 

7. 

* 

4' 

8. 

3 

7 

A 

I 
1 

9. 

«i 


1  5.    1.  8.    1.  11.    9. 

I  *  A  i* 

I.  6.  1  9.  ?i.  12.  It. 

HI 

3l~lj  21     3£  of  4^ 
2|-|  •   2iof6^ 

.   7A-4A  22.     §  +  2i 
5A-2f  1A  +  2* 


2  _i_   1 

i  +  i 

tt-t 

tt  +  * 

i  +  A 

i  +  A 

A- A 

15.    U.H.  19.    15A-1°^  23.     3*  +  4^ 

18f-16A  6j  +  lTv 

20.    fLzll.  24.        i  +  *+£ 

I  °i  I  A+t**+A 


A  ~"TT 

14    1    18 

-A-l 

1    4-  1  — 
TO"  tf 

■A-A 

29. 


1 


2- 


22  3  + 


26.    — *V*  2_i 

1  +  8Tf  30.    * 


1  + 


27.    —^—. 


5- 


7-f 


31. 


3  +  4 

7 


28-2^S- 


5     3i  +  2J  3-f 


Art.  127.]                COMPLEX  FRACTIONS.  107 

32.    3 33.    ?& . 

Q    I  jj  Ql jj 


127.  We  now  proceed  to  give  examples  of  a  more 
complicated  nature;  it  will  be  well,  however,  for  the 
student  to  consider  carefully  the  following  cases  in  which 
mistakes  are  frequently  made  in  the  meaning  of  the  signs 
employed. 

I.  Operations  of  multiplication  and  division  are  to  be 
performed  in  order  from  left  to  right,  and  each  sign  is  a 
direction  to  multiply  or  divide  what  precedes  the  sign  by 
the  number  that  follows  next  after  it. 

For  example,  36  x  6  +  3  =  216  +  3  =  72, 


36 -6-3  =  6-3  =  2, 

and 

36  -=-  6  x  3  =  6  x  3  =  18. 

So  also, 

2X3.    8_2x3  .    8  _  2  x  3      16  . 
3     5  *  15     3  x  5      15     3x5      8 " 
2.5. 4_2      6 . 4_2 v6      5      , 
3     6      5~35  *  5~3X5     4       ' 

_3 
"4' 

and 

2  .  6     4_2     6     4      16 

3  "  6     5      3     5      5     25 

II.  Numbers  connected  by  the  sign  '  of '  must  be  con- 
sidered as  a  single  number,  just  as  if  they  were  enclosed 
in  brackets. 

Thus,    M^of7  =  U^2_^  =  Ux52^  =  8> 
15      5       8      15      5x8      15     2x7      3 

Again,?of^§of^  =  ^^^§^  =  3-^><§JL6  =  §. 
8        4       8      8       6     4x8      8x6      4x8      3x5      2 

III.  Before  performing  any  operations  of  addition  or 
subtraction,  all  multiplications  and  divisions  must  be 
performed,  and  complex  and  compound  fractions  must 
be  reduced  to  simple  fractions. 


108  FRACTIONS.  [Chap.  IV. 

Thus,MofM  =  2  +  ^  +  § 
'3     4       6^4     3^4x6^4 

=  ?,      6      ■  3_16+15  +  18  =  49_Q1 

3"1"     8     +  4  24  24        *¥' 

It  is  a  very  common  mistake  to  work  a  question  of  this  kind  as 
ifitmeaat  +      of      + 


EXAMPLES    XXXIX. 
Written  Exercises. 


12 


Simplify : 

1.  fxfxf  15.   284f  of  }f -h  17J 

2.  i  +  *Xf  16.   ^+l|«f4J«rflf 
3-  l-f^-f  17-   lof|^13Joff 

4.  |  +  A+f  18-   4foflf  +  4£of2f 

5.  H+i*2i>  19.  ty+fafft  +  q. 

6.  1J  +  JX2J.  20.  li  +  SJoff  x6J. 
7-  }+|xA+f  21.  2j-lfof  1^  +  4 

s-  A+fxA+q.  22-  *  +  i«fi-f 

9.  6Jx4t-f.q-r.2f.  23.  |of|  +  |-|. 

10.  |  +  4of^.  24.  ^-Jofl-J. 

11.  frff-l-A-  25.  |ofi-|off 

12.  f«rfq-»-f  26.  2|  +  1\  of  2£  -  3J . 

13.  ^of3f-=-2f  27.  2£  of  q  +  21,  of  3f 

i4.  q-t-ifofq  28.  3|-Aof2j-if 

29.   fof3f-2|of  ^of2^-. 

so.  liofi+q-A- 

31.   f  of  2f  -  4|  of  5£ -=- 2|  of  3}. 


Art.  128.] "  EXAMPLES.  109 

32.  f  +  U<**t-i-**ifr- 

33.    3loflTV  +  7i-l|-|of^. 

35.    2i-|°flJ 
iof3i  +  if 

'    »  +  *«        2^i 

37      *  +  »  +  &   .T5IQf2i 

*       i-TV      '   *  at  4* 
21-jofll  +  f 


39. 


128.  To  express  one  number  or  quantity  as  a  fraction  of 
another,  we  proceed  as  follows : 

Ex.  1.   Express  174  as  a  fraction  o/188. 

Now  1  =  Tis  of  188  ; 

...  174  -  i%±  of  188 
sfg    of  188. 

Ex.  2.   Express  2\  dollars  as  a  fraction  0/8  dollars. 
Now  1  dollar  =  $    of  8  dollars  ; 

.-.  2 1  dollars  =  ?f  of  8  dollars 
=  T\  of  8  dollars. 

That  number  or  quantity  which  is  the  part  must  be  the  numerator, 
while  the  other  number  must  be  the  denominator,  of  the  required 
fraction. 


HO  FRACTIONS.  [Chap.  IV. 

EXAMPLES   XL. 
Oral  Exercises. 

1.  Express  27  as  a  fraction  of  81. 

2.  Express  140  pounds  as  a  fraction  of  280  pounds. 
What  fraction  of 

3.  9    is  3?  6.   49  is  7?  9.    9    is2j? 

4.  11  is  7?  7.   56  is  49?         10.    16  is  2$  ? 

5.  20  is  5?  8.   88  is  4?  11.   2iisT\? 

Written  Exercises. 

12.  How  many  times  does  8 J  feet  contain  2 J  feet  ? 

13.  Express  -£$  of  4  dollars  as  a  part  of  7  dollars. 

14.  Eeduce  2\  of  11  cents  to  the  fraction  of  5f  of  15 
cents. 

15.  What  would  be  the  measure  of  |-  of  23  tons,  if  \  of 
4  tons  were  used  as  the  unit  ? 

16.  If  the  income  of  A  is  -f-  of  f  of  1260  dollars,  and  the 
income  of  B  is  T8T  of  g1^-  of  5440  dollars,  how  large  is  A's 
income  compared  with  B's  ?  How  large  is  B's  income 
compared  with  A's  ? 

What  fraction  of 

17.  (8-2T3)(6+ 7-32)is23? 

18.   16  X  17         is  4[6  -  jll  -(3  +  U)  j  +  2]  ? 

K20|-6|-2)  2"        J 

1Q    488x11—1  of  7500  .    fttl+MS     2    ,,,     .  55- -= 
3*x5+l is  (6+14) -|  of  J/xi 22-5. 

129.   Reduction  of  Decimals  to  Common  Fractions. 

Decimals  may  be  considered  as  fractions  with  powers 
of  10  for  denominators. 

Thus,  .6  =  ^;     M  =  J&;     .002007  =  T&°&W 


Arts.  129,  130.]     REDUCTION  TO  DECIMALS.  HI 

Ex.  1.  Beduce  .76  to  a  common  fraction. 

7«  _     7  6     —19 

Ex.  2.   Beduce  4.012  to  a  mixed  number. 

4.012  =  4  -f  THo  =  M o- 

130.   Reduction  of  Common  Fractions  to  Decimals. 

Ex.  1.   Express  /5  as  a  decimal. 

Since  /s  may  be  considered  as  the  quotient  obtained  by  dividing 
4  by  25,  we  have  only  to  perform  this  division.     Thus, 

25)4.00(.16 
2  5 
1  50 
1  50 

Ex.  2.  Beduce,  to  3  places  of  decimals,  the  common  fractions, 
|,  £},  <md  f ;  and  thus  show  that  the  fractions  are  in  ascending 
order  of  magnitude. 

The  decimals  required  are  .75,  .854...,  and  .857.... 


EXAMPLES  XLI. 

Oral  Exercises. 

leduce 

i  to  decimals : 

i.  i. 

4.    f. 

7.    f. 

10. 

f 

2.    J, 

5.    f. 

8.    »?> 

11. 

t- 

3.    |. 

6.    1 

9.    f. 

12. 

H- 

Written  Exercises. 


13-  ft. 

16. 

**&• 

19.    £V        22-    7TVA- 

I*-  m 

17. 

T2"T' 

20.   ^.         23.    W^yy^- 

15-  tIt 

18. 

21 
l2~0"' 

21-   3»Hr 

112  FRACTIONS.  [Chap.  IV. 

Circulating  Decimals. 

131.  We  have  hitherto  considered  examples  of  division 
of  decimals  in  which  by  proceeding  far  enough  an  exact 
quotient  is  found  with  no  remainder.  This,  however,  is 
by  no  means  always  the  case ;  in  fact,  it  is  very  rarely 
the  case. 

Consider,  for  example,  the  division  of  5  by  3. 
3 1 5.0000 
1.6666... 
We  may  here  continue  the  process  of  division  to  any  extent,  but 
each  figure  of  the  quotient  will  be_6,  and  the  remainder  will  always 
be  2. 

Again,  divide  2  by  7. 

7  1 2.000000000 
.285714285... 
Here  the  six  digits,  2,  8,  5,  7,  1,  4,  come  over  and  over  again  in 
the  same  order,  and  we  shall  never  arrive  at  a  stage  at  which  there 
is  no  remainder. 

When  a  decimal  ends  with  digits  which  are  repeated 
over  and  over  again  without  end  in  the  same  order,  the 
decimal  is  called  a  Recurring  or  Circulating  decimal,  and 
the  digit,  or  set  of  digits,  which  is  repeated,  is  called  the 
Circulating  Period,  called  also  the  Repetend. 

Thus,  2.45555...,  .014141414...,  and  5.1246246246...  are  circu- 
lating decimals  with  circulating  periods  of  one,  two,  and  three 
figures,  respectively. 

A  circulating  period  is  denoted  by  placing  dots  over 
the  first  and  last  of  the  figures  which  recur. 

Thus,  2.45  denotes  2.45555...,  .014  denotes  .014141414...,  and 
5.1246  denotes  5.1246246246... 

A  circulating  decimal  is  said  to  be  Pure  or  Mixed, 
according  as  all  the  figures  after  the  decimal  point  do  or 
do  not  recur. 


Art.  131.]  EXAMPLES.  113 

Thus,  5.6,  31.24,  and  14.i35  sue  pure  circulating  decimals;  and 
.56,  3.124,  and  .14135  are  mixed  circulating  decimals. 

A  decimal  which  contains  a  definite  number  of  figures 
is  called  a  Terminating  decimal,  to  distinguish  it  from  a 
circulating  decimal,  which  contains  an  unlimited  number 
of  figures. 

Note.  Although  it  is  not  possible  to  reduce  any  common  frac- 
tion to  a  terminating  decimal,  it  is  always  possible  to  find  a  decimal 
which  is  equal  to  the  common  fraction  to  any  degree  of  accuracy 
that  may  be  required. 

For  example,  $  lies  between  .333  and  .334,  so  that  the  difference 
between  £  and  .333  is  less  than  one  one-thousandth,  so  also  the 
difference  between  ^  and  .333333  is  less  than  one  one-millionth; 
and  so  on. 

Now  there  is  no  species  of  magnitude  which  can  be  measured 
with  perfect  accuracy.  It  would,  for  instance,  be  difficult  to  deter- 
mine the  length  or  the  weight  of  a  body  without  a  possible  error  as 
great  as  one  one-thousandth  of  the  whole.  Hence  the  measure  of 
any  quantity  can  be  expressed  as  accurately  by  means  of  decimals 
as  by  means  of  fractions. 

EXAMPLES  XLII. 
Written  Exercises. 

Express  the  following  quotients  as  circulating  decimals : 
1.   1.5-2.7.  4.    .035 -.072.  7.   3.1-7. 


10  -  .03. 

5. 

.316  -r-  2.4. 

8.    15.6 -.07. 

1.7  -  .09. 

6. 

.312  -  8.8. 

9.    1.25-13.2. 

0.   5.193 -r- 

.0168. 

13. 

.3157  -  .259. 

1.   .0235 -^ 

.00616. 

14. 

27.31  -  6.475. 

2.   16.72- 

.0143. 

15. 

693.11  -  .011396. 

Reduce  the  following  common  fractions  to  circulating 
decimals : 

16.  |.  18.    f.  20.    TV  22.    &. 

17.  f  19.    ^  21.    ft.  23.    iV 


114  FRACTIONS.  [Chap.  IV. 

24.  2&.  26.    5Jf  28.    11^.  30.    2fif. 

25.  3if  27.    7^.  29.    13TV9,.       31.    5^V 

132.  Reduction  of  a  Circulating  Decimal  to  an  Equivalent 
Common  Fraction. 

We  have  seen  (Art.  129)  that  a  terminating  decimal  can 
be  expressed  as  a  common  fraction.  We  have  now  to  show 
that  a  circulating  decimal  may  be  expressed  as  a  common 
fraction. 

Consider  the  decimals,  .31,  .5216,  and  .15607. 

In  each  case  multiply  the  decimal  by  that  power  of  10  which  will 
move  the  decimal  point  to  the  end  of  the  first  recurring  period  ;  also 
(if  necessary)  multiply  the  decimal  by  that  power  of  10  which  will 
move  the  decimal  point  to  the  beginning  of  the  first  recurring  period. 
Subtract  the  second  product  from  the  first,  and  notice  the  result. 

0) 


.31 

X 

100 

=  31.31 

.31 

X 

1 

=      .3i 

.3i 

X 

99 

=  31. 

.-. 

.31 

_31 
99* 

No  advantage  will  be  gained  by  repeating  the  .31  in  the  minuend 
or  subtrahend ;  we  obtain  only  an  integer  in  the  remainder. 

(ii)         .5216  x  10000  =  5216.5216 
.5216  x  1  =  .5216* 


.5216  x    9999  =  5216. 

.-.  .5216  =  ffi!- 

(hi)  .15607  x  100000  =  15607.607 
.15607  x   100=   15.607 


.15607  x  99900  =  15592. 
.-.  .15607  =  iUW 


Art.  132.]  CIRCULATING   DECIMALS.  H5 

Three  facts  concerning  the  fraction  equivalent  to  a  cir- 
culating decimal  are  easily  noted : 

1.  The  numerator  is  the  whole  decimal  minus  the  number 
expressed  by  the  non-recurring  digits. 

2.  The  number  of  9's  in  the  denominator  equals  the 
number  of  recurring  digits. 

3.  The  number  of  naughts  in  the  denominator  equals  the 
number  of  non-recurring  digits. 

Ex.1.   i  =  f,  Ex.3.    .156=  #*  =  *■ 

Ex.  2.   .07  =  ft.  Ex.  4.    3.3L2  =  3|ff  =  8||f. 

It  should  be  noticed  that  by  the  above  rule  .9  =  f  =  1.  This 
result  can  be  seen  independently  ;  for  the  differences  between  1  and 
the  decimals,  .9,  .99,  .999,  etc.,  are  respectively  .1,  .01,  .001,  etc., 
each  difference  being  one-tenth  of  the  preceding,  and  therefore 
when  a  large  number  of  nines  is  taken,  the  difference  between  1  and 
.99999...  becomes  inconceivably  small. 

Since  .9  =  1,  .09  =  .1,  .009  =  .01,  and  so  on,  a  recurring  9 
can  always  be  replaced  by  1  in  the  next  place  to  the  left ;  for 
example,  .79  =  .8  and  .249  =  .25. 

EXAMPLES   XLIII. 
■Written  Exercises. 

Find  common  fractions  in  their  lowest  terms  equivalent 
to  the  following  circulating  decimals : 


1. 

.3. 

7. 

.185. 

13. 

.04878. 

2. 

.09. 

8. 

.396. 

14. 

.07317. 

3. 

17.27. 

9. 

.142857. 

15. 

9.23. 

4. 

.15. 

10. 

.285714. 

16. 

.79. 

5. 

1.027. 

11. 

.428571. 

17. 

6.36. 

6. 

.037. 

12. 

.012987. 

18. 

.315. 

116 


19. 

.116. 

20. 

.0254. 

21. 

.016. 

22. 

.749. 

FRACTIONS. 

[Chap.  IV. 

23.    .2027. 

27. 

11.3021976. 

24.    .19324. 

28. 

.542857L 

25.    .402439. 

29. 

.012345679. 

26.    .304878. 

30. 

.135802469. 

It  should  be  noticed  that  if  a  common  fraction  in  its 
loicest  terms  be  equivalent  to  a  terminating  decimal,  the 
denominator  of  the  fraction  can  contain  only  the  prime 
factors  2  and  5. 

133.  Addition,  Subtraction,  Multiplication,  and  Division 

of  circulating  decimals  are  performed  after  first  reducing 
to  common  fractions.  The  answer  in  each  case  should 
be  reduced  to  a  circulating  decimal. 

134.  An  exact  divisor  of  a  number  is  sometimes  called 
an  Aliquot  Part  of  the  number. 

2*  is  an  aliquot  part  of  10  ;  16|  is  an  aliquot  part  of  100. 

This  enables  us  to  use  a  short  process  of  multiplication 
(or  division)  in  cases  where  the  multiplier  (or  divisor)  is 
an  aliquot  part  of  some  power  of  10. 


Tox    31, 

x  10      and  - 

-3. 

To- 

8J,h 

-  10     and  x  3. 

To  x  12i, 

x  100    and  - 

=-8. 

To-f 

-  12i,  - 

r- 100    and  x  8. 

To  x  16|, 

x  100    and  - 

-6. 

To- 

-  16f , - 

-  100    and  x  6. 

To  x  25  , 

x  100    and  - 

-4. 

To- 

25  , - 

-  100    and  x  4. 

To  x  33|, 

X  100    and  - 

-3. 

To- 

83*,- 

- 100    and  x  3. 

To  x  125, 

x  1000  and  - 

-8. 

To- 

125,  - 

-  1000  and  x  8. 

Kead  the  signs  '  multiply  by '  and  '  divide  by '. 


32 


Square  Roots  of  Fractions. 

/3V=3  x3^ 
\±J      4x4     42 
it  follows  conversely  that 


135. 

Since 


4 


9  =3=  y9 
16     4     V16 


Arts.  133-135.]  SQUARE   ROOT.  H7 

Thus,  the  square  root  of  a  common  fraction  is  equal  to  a 
fraction  whose  numerator  and  denominator  are  respectively 
the  square  roots  of  the  numerator  and  denominator  of  the 
given  fraction. 

Ex.  1.    Find  the  square  roots  o/|||,  lT9g,  .4,  and  2.086419753. 
;144  _  y!44  _  12  .     V^JX=    /25  _  y25  _  5  , 
169      V169      13'  "Vm      V16     4' 


1144 

\169 


■Vs 


V9      3' 
and  V2.086419753  =  y/2fMtftffo  =  y/2% 

/169  _  yi69  =  13  =  i  4 
81       V81        9  ~    "  ' 
Ex.  2.  Find,  to  four  places  of  decimals, 

(i)  ^|     (")  ^     (iii)   V-3,    and    (iv)  JL 

(i)  ,»/-  =  -^--  =  -  V5*  which  can  be  fonnd  as  in  Art.  88. 

(ii)  In  examples  in  which  the  denominator  is  not  a  perfect 
square,  the  fraction  should  be  expressed  as  a  decimal.  In  the 
present  case  */-  =  ^.8  =  ...,  etc. 

Or  thus :  J| -J5  .Jflnlv»..-,  etc*. 

\5      \25      V25      6 

(iii)  .3  =  .33'33'33'33' ...     Then  proceed  as  in  Art.  88. 
(iv)  JL  =  _A*^L  =  6  v3  =  ...,  etc. 

The  change  of  form  from  —  to  -  V3  will  save  labor. 
V3      3 

EXAMPLES  XLIV. 
"Written  Exercises. 

Find  the  square  roots  of 


1     -Mr 

3.    iff. 

5.    39^.            7. 

.004. 

o       1    2  5 

4-    lift 

6.    .1.                   8. 

.134. 

9. 

1.361. 

10.    4.38204. 

118  FRACTIONS.  [Chap.  IV. 

Find,  to  four  places  of  decimals,  the  square  roots  of 

11.    tV.  13.  3£.  15.    2.4.  17.    .083. 

18.    3.5i62. 


12.    TV 

14. 

8f 

16.    .041. 

136. 

The  H.C.F. 

and  L.C.M 

.  of  Fractions. 

By  the  H.C.F.  of  two  or  more  fractions  we  mean  a  frac- 
tional H.C.F.     The  quotients,  however,  are  integral. 

A  fraction  -f-  a  fraction  =  an  integer  only  when  the  numerator 
and  denominator  of  the  dividend  divided  by  the  numerator  and 
denominator  of  the  divisor  respectively,  produce  an  integer  and 
the  reciprocal  of  an  integer  ;  thus, 

27      81      } 

Here,  14-4-2  is  an  integer,  and  27  -r-  81  is  the  reciprocal  of  an 
integer ;  i.  e. ,  the  numerator  of  the  divisor  is  a  factor  and  the 
denominator  of  the  divisor  is  a  multiple ;  also,  the  numerator  of 
the  dividend  is  a  multiple,  and  the  denominator  of  the  dividend  is 
a  factor. 

Hence,  the  H. C.F.  of  two  or  more  fractions  must  have  for  its 
numerator  the  H.  C.F.  of  the  given  numerators,  and  for  its  denom- 
inator the  L.C.M.  of  the  given  denominators. 

Also,  the  L.C.M.  of  two  or  more  fractions  must  have  for  its 
numerator  the  L.  C.  M.  of  the  given  numerators,  and  for  its  denom- 
inator the  H.C.F.  of  the  given  denominators. 

Note.  Before  obtaining  the  H.C.F.  or  the  L.C.M.,  the  given 
fractions  must  be  in  their  lowest  terms,  and  mixed  numbers 
must  be  reduced  to  improper  fractions.  The  L.C.M.  may  be 
integral. 

Ex.  1. 


Ex.  2. 


f  The  H.C.F.  of  f  and  i|  =  &  ; 
\  The  L.C.M.  of  f  and  J|  =  -5/. 

The  H.C.F.  of  |  and  §  =  ft  ; 

The  L.C.M.  of  f  and  T%  =  5, 


Art.  136.]  EXAMPLES.  119 

EXAMPLES    XLV. 
Written  Exercises. 

Find  H.C.F.  and  L.C.M.  of 

1.  |,2V,  and -y,  4.  Jf  ||,  and  ||, 

2.  A>665>andLf.  5.    fftandftf 

3.  ]»-&{>  and  ft  6.   ff  and  Iff 

7.   A85.  t6A>  and  39 &. 

EXAMPLES    XLVI. 
Miscellaneous  Examples.    Chap.  IV. 

1.  Reduce  5-f,  8y3T,  and  25^-  to  improper  fractions. 

2.  Simplify  |_3+_5___5_  +  _7__  _7_ 

3.  What  must' be  added  to  5J  that  the   sum   may 
bel2f? 


4.   Multiply  2|  of  5f  by  3|  -  6f 

3       1     1    nf   S  5  ' 

T4  -h  3  0I  ¥  —  2T 


31    _   5    v    61       3 

5.    Simplify  ?*  f  T4 


6.  Arrange,  in  ascending   order   of  magnitude,  the 
fractions,  T\,  f,  T85,  ft 

7.  From  the  sum  of  %  and  i  take  the  difference  be- 
tween \  and  |. 

8.  Simplify  2f  of  4§  of  5f 

9.  Simplify  3^  +  2^  7  4»x  A. 

12  ~f 

10.  What  fraction  of  350  equals  f  of  168  ? 

11.  Reduce  ft$$f  and  ||}||  to  their  lowest  terms. 

12.  Eeduce  to  a  common  denominator  ^¥,  g-2-^-,  T^-S} 
and  T|^. 


120  FRACTIONS.  [Chap.  IV. 

13.  Simplify  3|  +  2f  of  1$  -  4$. 

14.  Simplify  t'ttt"1?- 

15.  A  and  B  started  on  a  tour  with  192  and  156  dollars 
respectively,  and  they  had  equal  sums  left  at  the  end. 
A  spent  J  of  his  money ;  what  fraction  did  B  spend  of  his  ? 


16.  Add  TV,  &>  A>  lWr>  A7o>  and  ^V 

17.  Subtract  5Jf  from  7^ ;  also,  6f  +  2f  from  12 J. 

18.  Divide2T^4-2||-3/Tby2iV  +  3i-4f 
19  simplify  l*  +  2jaf5t-12t> 

20.   What  is  the  value  of  f  of  a  property,  if  -f  of  it  is 
worth  750  dollars  ? 


21.  Eeduce  ffjf,  £ft£M,  and  i^ff*  to  their  lowest 
terms. 

22.  Show  that  1  +  - t  +  ^— 5 — ; — =—7:  is  less 

2     2x3x4     2x3x4x5x6 

than  ^-,  but  greater  than  T\. 

23.  Simplify  fff  X  f f  X  & 
2j  +  1  of  lj  -  34- 

3    _|_    1    _;_   2   _   1_ 

1"     I     4     *     3"  8 

25.  Find  the  G.C.M.  of  5J  and  4|;  express  the  answer 
as  a  circulating  decimal  and  obtain  the  square  root. 

26.  Simplify  4  +  |  +  f  +  f  -  ¥  ~  2V 

27.  Subtract  2|f  from  5£f,  and  8if  from  12^. 

28.  Multiply  3£  of  5|  by  4-J-  of  f ,  and  divide  the  result 
by  41  of  1J. 


24.    Simplify  ^3~Vi   ZT     l 

4"     1     4  "*"  3"  ~~  ¥ 


Art.  136.]  MISCELLANEOUS  EXAMPLES.  121 

29.  Sunphfr  f^^Vi^f  of  jL_. 

30.  Find  the  L.C.M.  of  J  and  §. 

31.  By  how  much  does  the  sum  of  1^,  f,  and  ^  fall 
short  of  the  sum  of  -J,  |-,  J,  and  ii  ? 

32.  Simplify  1\  -  \  of  4J  +  2|  +  3*  X  2J  -  ^. 

33.  How  many  pieces  each  f  of  1  inch  can  be  cut  from 
a  wire  whose  length  is  5]  inches ;  and  what  will  be  the 
length  of  the  piece  left  over  ? 

34.  Simplify  ^  -  3j  +  5tV  _llf  -  5Ty 

35.  Find  L.C.M.  and  H.C.F.  of  ^fo  |fj,  and  T|§v 


36.  Take  the  sum  of  f  and  \  from  the  sum  of  £  and  f . 

37.  After  taking  away  \  and  f  of  a  certain  quantity, 
what  fraction  of  the  whole  will  be  left  ? 

38.  Multiply  \\  +  3|  by  3J  +  2f,  and  divide  the  result 
by  51  of  5|. 

39    Simplify  2H  ~  A  of  4Ty  +  3^  +  7f 
39'    SimpM>       3AoflOTV-3i-^4t     * 

40.   Find  the  value  of  3T3T  of  4f  of  If  of  Jj, 


41.  Add  |,  f|-,  -J^-,  TT0-,  and  283o6. 

5 

42.  Simplify  — 

91  _2  ftf15 

43.  Simplify  ?      '/V,',  and 

T  OI  °  3"  T  "8 


i L 


2  +  -!- 


122  FRACTIONS.  [Chap.  IV. 

44.  There  are  three  partners  in  a  certain  business,  one 
of  whom  provided  -f  of  the  whole  capital,  and  another 
provided  f.  What  fraction  of  the  whole  was  supplied 
by  the  third  partner  ? 

45.  A  man  gives  \  of  his  money  to  his  wife,  \  of  the 
remainder  to  his  son,  and  \  of  what  then  remains  to  his 
daughter ;  and  has  still  left  a  sum  of  1350  dollars.  How 
much  was  there  at  first  ? 

1 


1- 


46. 


Simplify  9i9^"689  ,77tand r^ 


2  +  i 


47.   Divide  1£  of  5|  by  2f  of  7^-. 

»-^£»-» 

49.  Find  the  value  of  f  of  f  of  5  dollars  -  1  of  £  of  2 
dollars,  and  express  the  difference  as  a  fraction  of  11.25 
dollars. 


50.   Reduce  to  its  simplest  form 

(3t  +  5|-A-)(4i-3i) 


lA  +  2*-(2A-i-A) 

51.  After  spending  J-  of  his  money,  a  man  found  that 
f-  of  the  remainder  was  63  cents ;  how  many  cents  had  he 
at  first? 

52.  I  purchased  some  square  tiles  for  a  room  483 
inches  long  and  266  inches  broad ;  the  manufacturer  sent 
me  the  largest  tiles  I  could  use ;   how  long  was  each  tile  ? 

53.  A  man  travelled  f  of  a  certain  distance  by  railway, 
-^  of  the  whole  distance  by  coach,  and  walked  the  rest 
of  the  way,  which  was  15  miles.  What  was  the  length  of 
the  whole  journey  ? 


Art.  136.]         MISCELLANEOUS   EXAMPLES.  123 

54.  By  what  must  7§  of  3|  be  multiplied  that  the  prod- 
uct may  equal  4f  of  2f  ? 

55.  Simplify     g*I^M^M±lli;    ^^  ^ 

answer  into  two  fractions  so  that  one  factor  shall  be  a 
perfect  square. 

56.  Find  the  H.C.F.  and  the  L.C.M.  of  ££  and  f  f 

57.  Subtract  10ft  from  23|,  and  16|f  from  20| J. 

58.  Simplify  (3*  +  21  +  4J)  +  (J  +  f)  of  (|  -  *)• 

59.  Simplify 

Q  2 
3  _  2    v   I   —  1  °T¥ 

U   (4i-2|)  +  (3i-li)'   <■  ;  2J-J-4 

4—  4 

60.  A  man  gives  away  f  of  his  money  and  afterwards 
-J  of  the  remainder.  What  fraction  of  the  whole  had  he 
then  left? 

61.  Reduce  to  a  common  denominator,  and  arrange  in 
order  of  magnitude  the  fractions,  -j^-,  T7^,  -g9^,  if?  if • 

62.  Multiply  the  difference  between  3£  of  1TV  +  7| 
and  2\  -+-  f  of  -^  by  the  sum  of  _  and  — 

63.  Simplify  I  of  69~2f^   K¥  +  19t). 

64.  After  spending  -|  of  his  money,  a  boy  found  that  $ 
of  the  remainder  was  2|  dollars.     What  had  he  at  first  ? 

65.  Reduce    to    their   lowest   terms    f^J,   y^ff,   and 

66.  A  gave  i  of  his  marbles  to  B,  i  to  C,  i  to  D,  Jg- 
to  E,  and  then  had  105  left.  How  many  did  each 
receive  ? 


124  FRACTIONS.  [Chaps.  IV.,  V. 

67.   Find  2||of  {2J|  -4<rf  (3i  of  2f  -  5|)  of  1J  -  Af. 


68.  64  feet  of  brass  rods  cost  121  cents  a  foot;  what 
was  the  cost  of  the  rods  ? 

69.  Nashville,  Tenn.,  Jan.  1,  1895. 
J.  S.  Cushing  &  Co.  To  H.  A.  Armstrong,  Dr. 


For    2  lb.  Sugar  @     7  cents 

"      5   "    Tea       "50     " 

"    11    "    Coffee    "34      " 

"    17   "    Starch  "14      " 

Find  the  amount  of  the  above  bill ;  answer  in  dollars 
and  cents,  letting  100  cents  equal  one  dollar. 

What  would  be  the  answer  in  dollars,  and  the  decimal 
of  a  dollar  ? 

^    70.   Find  the  H.C.F.  and  L.C.M.  of  1.485  and  12.6. 


71.    Simplify  m*< 


81x32x34 

72.  Add,  without  changing  positions:  67.04,  12,  5^, 
9T\,  4.17,  243^,  14,  ST\. 

73.  A  certain  lake  is  .327  of  a  mile  long;  what  is  its 
length  compared  with  the  length  of  a  second  lake  2|  miles 
long? 

(Answer  must  be  reduced  to  a  circulating  decimal.) 

74.  Three  tanks  contain  924,  1500,  and  2520  gallons  of 
water  respectively ;  what  is  the  largest  number  of  gallons 
that  can  run  from  each  of  the  tanks  per  minute  and  allow 
all  to  be  emptied  in  a  whole  number  of  minutes,  the  rate 
of  flow  from  each  tank  being  the  same?  How  many 
minutes  are  required  to  empty  each  tank  ? 


•  I 


Arts.  137,  138.]     MAGNITUDE  —  QUANTITY.  125 


CHAPTER  V. 

DECIMAL   MEASURES. 

137.  Anything  which,  can  be  increased  or  diminished 
is  called  a  Magnitude. 

Lengths,  areas,  weights,  etc.,  are  magnitudes. 

To  measure  a  magnitude  is  to  compare  it  with  some 
known  magnitude  of  the  same  kind,  which  is  taken  as 
a  unit,  and  to  say  how  many  times  the  unit  must  be 
repeated  in  order  to  make  up  the  magnitude  in  question. 

For  example,  to  measure  any  given  length  of  string,  is  to  find 
how  many  times  some  known  length,  say  a  foot,  must  be  repeated 
to  make  up  the  given  length ;  and  this  number  of  times  is  called 
the  measure  of  the  length. 

A  measured  magnitude  is  called  a  Quantity. 
Thus,  any  quantity  is  expressed  by  a  number  and  a 
unit  of  the  same  kind  as  itself. 

138.  Numbers  are  first  used  in  connection  with  distinct 
objects,  and  are  afterwards  used  in  measuring  continuous 
magnitudes  of  any  kind.  If  the  continuous  magnitude 
cannot  be  measured  by  one  unit,  a  series  of  units  smaller 
and  smaller  in  value  may  be  used. 

For  example,  to  measure  a  string,  some  definite  length,  say  a 
yard,  is  fixed  on  as  a  unit.  Suppose  the  given  string  contains 
Q\  yards.     We  may  use  a  second  unit,  say  a  foot,  to  measure  the 


126  DECIMAL  MEASURES.  [Chap.  V. 

^  yard.  If  there  are  3  feet  in  one  yard,  the  \  yard  will  be  \\  feet, 
and  the  string  will  measure  6  yards  \\  feet.  This  \  foot  may  be 
expressed  in  a  smaller  unit  still,  say  an  inch  ;  if  there  are  12  inches 
in  a  foot,  the  \  foot  will  be  6  inches,  and  the  string  will  measure 
6  yards  1  foot  6  inches. 

139.  Quantities  expressed  in  terms  of  a  single  unit 
are  called  Simple  Quantities,  and  quantities  which  are 
expressed  in  terms  of  more  than  one  unit  are  called 
Compound  Quantities. 

To  measure  every  different  kind  of  quantity,  some 
standard  unit  is  employed,  and  also  other  units  which 
are  obtained  by  subdivisions  and  repetitions  of  the 
standard  unit. 

Units  which  require  10  of  one  to  make  one  of  the  next 
higher  are  the  simplest  to  use.  Such  units  are  called 
Decimal  Units. 

In  numeration  of  quantities,  units  of  different  kinds 
are  called  units  of  different  denominations. 


Tables  of  Decimal  Units. 

140*   Table  of  United  States  Money. 
Money  is  a  measure  of  values. 

10  mills  (m.)  =  1  cent  (ct.). 
10  cts.  =  1  dime  (d.). 

10  d.  =  1  dollar  ($). 

10  $  =  1  eagle  (e.). 

The  eagle  is  usually  called  ten  dollars,  and  the  dime  is 
usually  called  ten  cents;  so  that  the  only  names  generally 
used  are  dollars  and  cents. 

*  It  is  advisable  to  study  numeration  and  notation  of  decimal 
measures  at  the  same  time. 


Arts.  139-143.]  U.  S.  COINS.  127 

Thus,  $25.35  is  read,  '25  dollars  35  cents,'  and  not,  '2  eagles 
5  dollars  3  dimes  5  cents ' ;  also,  $.20  is  read,  '  20  cents.' 

The  notation  is  as  follows : 

The  figure  representing  eagles  is  put  in  tens'  place, 
u  "  dollars    "      "    units'  place, 

"  "  dimes     "      "    tenths'  place, 

"  u  cents       "      "    hundredths'  place, 

"  "  mills       "      "    thousandths'  place. 

141.  A  sum  of  money  represented  in  any  denomina- 
tion may  be  represented  in  higher  denominations  by 
moving  the  decimal  point  to  the  left,  one  place  for  each 
denomination.  A  reduction  is  made  to  lower  denomina- 
tions by  moving  the  decimal  point  to  the  right ;  thus, 

6742  mills     =  67.42  dimes  =  6.742  dollars  ; 
4671  dollars  =  46.71  cents  =  467.1  mills. 

142.  We  have  already  shown  how  to  perform  the 
operations  of  addition,  subtraction,  multiplication,  and 
division  of  decimals ;  and  the  application  of  these  rules 
to  sums  of  money  will  require  no  further  explanation, 
except  to  state  that  in  cases  of  addition  and  subtraction 
care  must  be  used  in  writing  units  of  the  same  denomina- 
tion in  the  same  vertical  column.  This  is  not  a  necessity 
—  only  a  convenience. 

143.  The  coins  in  use  are  as  follows : 

Gold  coins :  the  dollar,  the  quarter-eagle,  the  half-eagle, 
the  eagle,  and  the  double-eagle. 

Silver  coins:  the  dollar,  the  half-dollar,  the  quarter- 
dollar,  and  the  dime. 

Nickel  coin :   the  five-cent  piece. 

Bronze  coin:    the  cent. 

The  mill  is  used  only  in  computation. 


128  DECIMAL   MEASURES.  [Chap.  V. 

EXAMPLES   XLVII. 
Written   Exercises. 

1.  Write  the  following  in  figures :  two  dollars  thirteen 
cents,  sixty  dollars  forty  cents,  three  hundred  dollars  two 
cents,  sixteen  cents,  six  cents,  three  cents  five  mills. 

2.  Add  $14.15,  $37.24,  $156.50,  $.75,  and  $1204.06. 

3.  Add  $2.04,  26.7  ct.,  49.62  m.,  and  4.338  ct. 

4.  By  how  much  is  $1507.45  greater  than  $1429.78? 

5.  After  spending  $145.45  a  man  had  $13.55  left; 
how  much  had  he  at  first? 

6.  A  man  had  originally  $1345.40.  How  much  had 
he  left  after  paying  away  $135.25,  $416.67,  and  $575.48  ? 

7.  What  will  250  barrels  of  apples  cost  at  $2.75  per 
barrel  ? 

8.  A  man  bought  150  horses  at  $125  each.  He  sold 
50  at  $145  each  and  the  rest  at  $137.50  each.  How 
much  did  he  gain  ? 

9.  What  is  the  value  of  1400  bushels  of  wheat  at  67 
ct.  a  bushel  ? 

10.  A  man  bought  1250  bushels  of  oats  at  381  ct.  a 
bushel,  and  1500  bushels  of  wheat  at  65|-  ct.  a  bushel. 
What  was  the  whole  cost  ? 

11.  A  man  bought  15  pounds  of  cheese  at  $.14  a 
pound,  9  pounds  of  coffee  at  $.25  a  pound,  and  13 
pounds  of  butter  at  18  ct.  a  pound.  How  much  did  the 
whole  cost? 

12.  Two  men  had  between  them  $1595,  and  one  had 
$155  more  than  the  other.     How  much  had  each  ? 


Art.  144.]  THE   METRIC   SYSTEM.  129 

13.  Multiply  $684.93  by  6.75. 

(Give  answer  to  two  decimal  places,  remembering  that  5  or 
more  mills  increase  the  number  of  cents  by  one  ;  anything  under 
5  mills  is  not  considered.) 

14.  Multiply  $71.41  by  .23. 

15.  Divide  $5687.98  by  27.3. 

(Be  sure  in  the  answer  to  find  out  whether  or  not  the  mills  will 
be  as  many  as  five.) 

16.  A  man  spent  $4.86  in  buying  beef  at  $.09  per 
pound.     How  many  pounds  did  he  buy  ? 

17.  A  man  bought  wheat  at  64  ct.  a  bushel,  and  spent 
$736  altogether.     How  many  bushels  did  he  buy  ? 

18.  How  many  d.  in  $560.1  ? 

19.  How  many  e.  in  $270  ? 

20.  How  many  m.  in  $41.90  ? 

21.  How  many  ct.  in  86420  m.  ? 

22.  How  many  $  in  86420  m.  ? 

23.  Divide  784  d.  by  2.75,  and  write  the  answer  as  e. 

24.  Multiply  $76  by  .0025;  of  what  denomination 
is  the  answer  ? 

The  Metric  System. 

144.  In  almost  all  civilized  countries,  the  United 
States  and  England  being  unfortunately  exceptions,  the 
different  weights  and  measures  have  been  arranged  on 
the  decimal  system. 

In  France,  Belgium,  and  Switzerland  all  sums  of 
money  are  expressed  in  terms  of  the  Franc,  with  its  sub- 
unit  the  Centime  (y^-  of  a  franc).  In  Italy,  Spain, 
and  Greece  the  standard  unit  of  money  is  of  exactly 
the  same  value  as  the  franc,  but  is  called  by  different 
names. 


130  DECIMAL  MEASURES.  [Chap.  V. 

In  Germany  the  standard  nnit  is  the  Mark,  with  its 
sub-unit  the  Pfennig  (y^  of  a  mark). 

In  Austria  the  standard  unit  is  the  Gulden,  with  its 
sub-unit  the  Kreutzer  (T^-g-  of  a  gulden). 

In  the  United  States  the  Metric  System  is  used  in 
scientific  investigations  and  is  authorized  to  be  used  in 
the  Mint  and  Post  Ofhce. 

145.  In  the  Metric  System  of  weights  and  measures, 
the  fundamental  unit  is  called  a  Meter.  The  meter  is 
approximately  the  one  ten-millionth  part  of  the  distance 
from  the  equator  to  the  north  pole.  (A  slight  error  was 
made  in  obtaining  the  meter,  but  its  length  remains  as  at 
first  calculated.) 

The  standard  units  of  area,  volume,  capacity,  and  weight 
are  derived  from  the  meter. 

Decimal  divisions  of  a  standard  unit  are  distinguished 
by  the  Latin  prefixes  deci-,  centi-,  milli-. 

Decimal  multiples  of  a  standard  unit  are  distinguished 
by  the  Greek  prefixes  deka-,  liekto-,  kilo-,  myria-. 

Tables  of  Decimal  Units.  —  Continued. 

146.  Table  of  Linear  Measures. 

Length  is  distance  in  a  straight  line  between  two 
points. 

The  unit  of  linear  measure  is  a  meter. 

10  millimeters  (mm)  =  1  centimeter  (cm). 

10 cm  =1  decimeter  (dm). 

10dra  =1  meter  (m). 

10 m  =1  dekameter  (Dm). 

10 Dm  =1  hektometer  (Hm). 

10 Hm  =  1  kilometer  (Km). 

10 Km  =1  myriameter  (Mm). 


Arts.  145-148.] 


LINEAR   MEASURES. 


131 


If  the  figure  representing  meters  is  put  in  units'  place, 
then         "  "  dm        "       "     tenths'  place, 

"  "  cm         u       *«     hundredths'  place, 

"  "  Dm       "       "     tens'  place, 

"  "  Hm       "       "     hundreds'  place,  etc. 


147.  Length,   represented   in   any  denomination   may- 
be represented  in  higher  denominations  by  moving  the 
decimal  point  to  the  left,  one  place  for  each  denomina- 
tion;  a  reduction  is  made  to  lower   denomi- 
nations by  moving  the  decimal  point  to  the 
right;  thus, 

14.45  m  =144.5  d™  =  14450  "«■ ; 
1256.4  «n  -  12.564  ™  =  1.2564  »m  ; 
2  Km  _  200 Dm    =  20000  dm  ; 

24.6  Hm    =2.46Km   =  .246Mm. 

The  same  methods  are  used  in  operations  here  as 
in  decimals  [Arts.  29  and  49].  Units  of  the  same 
denomination  should  be  in  the  same  vertical  column. 

The  teacher  should  have  a  meter  stick,  properly- 
graduated,  and  keep  it  constantly  before  the  class. 
Not  a  word  should  be  said  about  any  other  linear 
measure,  known  or  unknown  to  the  class. 

148.  Figures  representing  decimal  measures 
of  any  kind  are  read  just  as  figures  represent- 
ing integral  and  decimal  numbers  are  read, 
and  then  the  name  of  the  denomination  repre- 
sented is  read ;  thus, 

14.45m  is  read  'fourteen  and  forty-five  hundredths 
meters, '  (which  means  the  same  as  if  it  were  read '  one 
dekameter  four  meters  four  decimeters  and  five 
centimeters'). 

.246  Km  is  read  'two  hundred  forty -six  thousandths 
kilometers.' 


1 

%      : 

S 

s* 

g 
o 
o 
i— i 

U 

II 
E 

o 
O 
t-H 

* 

II 

U 

a> 
I 

6      : 

o 
<x> 

t-H 

7         ; 

8      : 

9      '- 

10       ; 

132  DECIMAL   MEASURES.  [Chap.  V. 

EXAMPLES   XL VIII. 
"Written  Exercises. 

1 .  Cut  from  cardboard  a  narrow  strip,  ldm  long,  and 
mark  it  accurately  into  tenths  and  hundredths. 

2.  Obtain  the  measure  of  the  length  of  a  book,  and 
state  the  answer  in  dm  and  mm. 

3.  Mark   your   height   on  the  wall,  and   obtain  its 
measure  in  m ;  also  in  dm. 

4.  Measure  a  room  in  m;  obtain  length,  breadth,  and 
height. 

5.  Express  25 m  as  Dm;  as  Mm;  as  cm;  as  mm. 

6.  Write  126.73 Dra  as  m;  as  Km;  as  dm;  as  mm. 

7.  Add  14™,  6dm,  5027 mm,  and  6.5Bm.     Answer  in  m. 

8.  Find  the  number  of  Dm   in   12.62 Mm  +  4267 m  -f 
845 cm. 

9.  How  much  longer  is  a  room  12.65m  than  a  room 
106 dm  long? 

10.  Find  8469 ra + 46892 mm- 468 Dm+ 12 dm- 186 cm  in  m. 

11.  Multiply  78.6 dm  by  125.  Answer  in  m;  also  in 
Hm. 

12.  Four  m  of  ribbon  cost  16|  ct.  per  m;  find  total 
cost  [Theorem  I,  Art.  47 ;  also  Art.  50].  The  answer  is 
what  fractional  part  of  $1  ? 

13.  Divide  7469 mm  by  11.  Answer  in  three  denomi- 
nations. 

149.   Table  of  Surface  Measures  (Square  Measures). 
That  which  has  length  and  breadth,  but  no  thickness, 
is  called  a  Surface;  thus, 

The  surface  of  a  book  has  length  and  breadth. 


Arts.  149,  150.]         SQUARE  MEASURES. 


133 


A  portion  of  a  surface  bounded  by  lines  is  called  a 
Figure. 

A  plane  figure  bounded  by  four  equal  sides, 
and  whose  four  angles  are  equal,  is  called  a 
Square. 

Any  square  may  be  used  as  a  unit  of  surface 
for  instance,  a  square  centimeter,  or  a  square 


Square 
Centimeter. 


measure 
meter. 


100  square  millimeters  (qmm)  =  1  sq.  centimeter  (qcm). 
100qcm  =  1    "   decimeter  (qdm). 

100qdm  =  1    "   meter  (qm). 

100 qm  =  1    "   dekameter  (qDm). 

100  qDm  =1    "   hektometer(qHm). 

100*Hm  =1    "   kilometer  (qKm). 


For  Land  Surveying. 
lqm    is  called  a  centar  (ca). 

lqD»      U  U  an  ^   ^ 

lqHm     U  «         a  liektar  (Ha)# 

Sq.  cm,  etc.,  are  often  used  instead  of  qcm,  etc. 


150.   It  is  evident  from  the  figure  that,  if  one  square 


10  units 

lo 

ngi 

I  unit  long. 

□ 


is  10  times  as  long  as 
another,  its  surface  is  100 
times  as  large  j  there- 
fore, 

A  surface  represented 
in  any  denomination  may 
be  represented  in  higher 
denominations  by  moving 
the  decimal  point  to  the 
left,  two  places  for  each 
denomination;   a  reduction  is  made  to  lower  denomina- 


134  DECIMAL   MEASURES.  [Chap.  V. 

tions  by  removing  the  decimal  point  to  the   right,  two 
decimal  places  for  each  denomination;  thus, 

15.6<im    =  .156<iDm  ;  10625<imm  =  1.0625 idm  ;  12a  =  .12Ha. 
1.49qdm  =  i49qcm  .  i.0625<iHm  =  10625 im  ;    1.6a  =  160ca. 

15.6  <Jm  is  read  '  fifteen  and  six-tenths  square  meters.' 

EXAMPLES  XLIX. 
"Written  Exercises. 

1.  Cut  from  cardboard  a  piece  to  represent  one  qdm, 
and  mark  it  accurately  into  qcm. 

2.  How  many  qcm  in  \  a  qdm  ?     How  many  in  the 
square  of  ^  a  dm  ? 

3.  Mark  out  on  the  floor  a  qm.     What  would  that  be 
called  if  it  were  marked  on  the  ground  ? 

4.  Write  30 qm  15qdm  21qcm  as  qm ;  as  qHm ;  as  qmm. 

5.  Express  31qm  as  qcm;  14.1qm  as  qdm ;  .5qm  as  qcm; 
120.7 qKm  as  qm. 

6.  Read  15.14 qDm  as  qm ;   as  qcm.     Read  l.lqHm  as 
qDm;  as  qmm.     Read  121 a  as  ca;  as  Ha. 

7.  Multiply  78.141qm  by  16,  and  answer  in  qmm. 

8.  Represent  15.6789qm  as  qmm  ;  140(icm  as  qHm. 

9.  In  1.49 qdm  what  might  the  .49  be  called  ? 

10.  Read  15.6 qm  as  qDm  and  qdm. 

11.  Represent  lqKm,  12qHm,  lqm,  4qmm  as  qm. 

12.  Read,  stating  the  number  of  units  of  each  denomi- 
nation represented,  167.08193qHm. 

13.  Divide  78965 qDm  by  5,  and  answer  in  qm. 

14.  Add  167qm,  200qKm,  18.67 qDm,  and  160003qmm. 


Arts.  151,  152.] 


CUBIC  MEASURES. 


135 


15.  From  12.6a  subtract  4ca. 

16.  A  piece  of  ground  containing  400 a  is  2000 dm  long; 
what  is  its  breadth  ? 

151.  Table  of  Volume  Measures  (Cubic  Measures). 
A  solid  bounded  by  six  equal  square  sur- 
faces is  called  a  Cube. 

Any  cube  may  be  used  as  a  unit  of  cubic 
measure ;  for  instance,  a  cubic  centimeter  or 
a  cubic  meter. 

1000  cubic  millimeters  (cmm)  =  1  cubic  centimeter  (ccm), 
1000 ccm  =  1  cubic  decimeter  (cdm), 

1000 cdra  =  1  cubic  meter  (cum). 

For  measuring  wood,  1  cubic  meter  is  called  a  ster  (st). 

152.  It  is  evident  from  the  figure  that,  if  one  ciibe  is 
10  times  as   long   as   an- 


A 

/ 

/— 

_j 

Cubic 
Centimeter. 


Si 


/   /    /    / 


-/- 


p^g 


m 


other,  its  volume  is  1000 
times  as  large;  there- 
fore, 

A  volume  represented  in 
any  denomination  may  be 
represented  in  higher  de- 
nominations by  moving 
the  decimal  point  to  the 
left,  three  places  for  each 
denomination;  a  reduction 
is  made  to  lower  denomi- 
nations by  moving  the 
decimal  point  to  the  right,  three  places  for  every  denomi- 
nation. 

Thus,   4678 cdm  =  4.678 cum  ;    8.67cum  =  8670000 ccm. 

4.678 cum  is  read  i  four  an(i  six  hundred  seventy-eight  thousandths 
cubic  meters,'  which  is  equivalent  to  saying  four  cubic  meters  and 
six  hundred  seventy-eight  cubic  decimeters. 


10  units  long. 


s 

unit  long. 


136 


DECIMAL   MEASUEES. 


[Chap.  V. 


EXAMPLES  L. 
Written  Exercises. 

1 .    Cut  from  wood  (or  rubber,  or  cork)  a  piece  to  rep- 
resent lccm. 


(Each  student  should  carry  in  his 
pocket  such  a  piece  of  wood,  so  that 
he  can  constantly  refer  to  it.) 

2.  Cut  from  bristol  board  a 
piece  shaped  like  the  figure, 
having  each  of  its  six  squares 
1*»  long.  Cut  nearly  through 
the  cardboard  in  places  repre- 
sented by  dotted  lines,  and 
make  small  flaps  as  shown. 
Such  pieces  can  be  made  into  cubes ;  a  little  mucilage  on 
the  flaps  will  keep  the  cubes  in  shape.  Flaps  should  be 
out  of  sight  in  the  finished  cubes. 

3.  Mark  the  sides  of  the  cube  into  qcm  and  learn 
how  many  ccm,  like  the  one  in  your  pocket,  would  be 
required  to  make  a  block  as  large  as  the  cardboard  cube. 

4.  How  many  ccm  in  J-  a  cdm?  How  many  ccm  in 
the  cube  of  i  a  dm;  i.e.,  in  a  cube  5cm  on  an  edge? 

5.  Write  105 cum  215 cdm  as  cum;  27cdmascum. 

6.  Eead  10.516cum  as  cdm;  as  cDm. 

7.  Eead  10067 cdm  as  st. 

8.  Kead  100601.41cum,  stating  the  number  of  units  of 
each  denomination  represented. 

9.  Add  14.1cmm,  14.1cdm,  and  14.1cDm. 

10.    Divide  14.4 ccm  by  12,  and  write  the  answer  as  ccm, 
and  as  cmm. 


Arts.  153,  154.]  LIQUID  —  WEIGHT.  137 

153.  Table  of  Volume  Measures  (Liquid  Measures). 
The  cubic  decimeter  is  used  as  the  unit  of  measure, 

and  is  called  a  Liter. 

10  milliliters  (ml)  =  1  centiliter  (cl). 
10cl  =  1  deciliter  (dl). 

lO41  =  1  liter  (1). 

101  =  1  dekaliter  (Dl). 

10 D1  =  lhektoliter  (HI). 

10m  =lkiloliter  (Kl). 

Comparing  the  above  table  with  the  one  in  Art.  151, 
we  find  that 

•J^cdm  __  -J^l 
1  ccm  __  1  ml 
-Jcum__  ^Kl 

EXAMPLES   LI. 
"Written  Exercises. 

1.  Add  4.5 ",  2dl,  47 cl,  and  673  ml. 

2.  Express  the  answer  to  Ex.  1  in  Kl,  HI,  and  cl. 

3.  How  many  liters  of  water  in  4cum? 

4.  Change  46.0949Kl  to  1;  to  ml;  to  dl;  to  ccm. 

5.  Multiply  .678 cl  by  2693;  express  the  answer  in 
cdm,  and  in  cum. 

154.  Table  of  Measures  of  Weight. 

The  attraction  which  the  earth  and  any  other  body  (on 
or  off  the  earth)  have  for  each  other  is  called  Gravity. 

The  amount  of  this  attraction  is  called  the  Weight  of 
the  body. 

The  weight  of  lccra  of  water  is  the  unit  of  weight,  and 
is  called  a  Gram. 


138  DECIMAL   MEASURES.        [Chaps.  V.,  VI. 

10  milligrams  (mg)=  1  centigram  (eg). 

10 cg  =1  decigram  (dg). 

10dg  =lgram(g). 

10*  =  1  dekagram  (Dg). 

10Dg  =  1  hektogram  (Hg). 

10Hg  =  1  kilogram  (Kg). 

10 Kg  =1  myriagram  (Mg). 

10Mg  =  1  quintal  (Q). 

10 Q  =  1  tonneau  (T). 

Observe,  in  the  case  of  water,  that 

lml  (=lccm)  weighs  lg; 
l1    (=lcdm)  weighs  lKg; 
1K1  (=  lcum)  weighs  1T. 

155.   Kilogram  is  called  Kilo.    Quintal  is  not  often  used. 

The  cubic  centimeter  of  water,  which,  is  used  as  the  standard 
unit,  must  be  distilled,  must  be  at  a  temperature  of  39.2°  F. 
(4°  C),  and  must  be  weighed  in  a  vacuum  at  the  level  of  the  sea. 

EXAMPLES  LII. 
Written  Exercises. 

1.  Eead  64.95 g  as  dg,  cg,  mg,  and  Mg. 

2.  Eead  1256 ng  as  Kg,  Q,  T,  and  g. 

3.  What  is  the  weight  of  lml  of  standard  water?  Of 
10 ml  ?  Of  lcl  ?  Of  10cl  ?  Of  3dl  ?  Of  31  ?  Of  1000ccm  ? 

4.  Iron  is  7.8  times  as  heavy  as  water ;  what  is  the 
volume  (in  cdm)  of  29.25 Kg  ?  What  is  the  weight  of  2cdm  ? 
Of  55ccm  ?  Of  7.2ccm  ?  Of  1.67 ccm ?  Of  125 cum  ? 

5.  Find  the  value  (in  grams)  of  4Kg  -  18dg  +  18g 
+  67.896 mg  -  126.73cg  +  4T  -  11.6Mg. 

6.  Gold  is  19.5  times  as  heavy  as  water;  what  is  the 
weight  of  lccm  ?     Of  one  cubic  meter  ? 


Arts.  155-157.]     NON-DECIMAL   MEASURES.  139 


CHAPTER  VI. 

NON-DECIMAL  MEASURES. 

156.  The  simplicity  of  calculations  when  using  decimal 
measures  is  due  to  the  facts  that  changes  can  be  easily 
made  from  one  denomination  to  another  by  moving  the 
decimal  point,  and  that  several  denominations  can  be 
expressed  together  in  one  set  of  figures. 

In  Non-Decimal  measures,  called  also  Denominate  num- 
bers and  Compound  Quantities,  a  variety  of  divisors  is 
used  in  the  different  tables  in  order  to  change  from  low 
denominations  to  higher  ones ;  also,  it  is  unusual  to  ex- 
press several  denominations  together  in  one  set  of  figures. 

For  example,  consider  the  case  of  the  string  mentioned  in  Art. 
138.  There,  12  inches  equal  1  foot,  and  3  feet  equal  1  yard ;  and 
the  length  of  the  string  must  he  expressed,  not  with  the  denom- 
inations together  in  one  set  of  figures,  hut  each  denomination 
separately,  —  6  yards,  1  foot,  6  inches. 

To  express  a  compound  (Art.  139)  quantity,  express  the  number 
of  units  of  each  denomination  separately,  indicating  the  denomi- 
nations, as  in  the  above  illustration. 

To  read  compound  quantities,  read  them  exactly  as  expressed. 

157.  Table  of  Measures  of  Time. 

The  Standard  Unit  of  Time  is  the  Mean  Solar  Day; 
that  is,  the  mean  interval  between  two  successive  pas- 
sages of  the  sun  across  the  meridian  of  any  place.  A 
day  is  supposed  to  begin  at  midnight. 


140  NON-DECIMAL  MEASURES.  [Chap.  VI. 

60  seconds  (sec.)  =  1  minute  (min.). 
60  min.  =  1  hour  (hr.). 

24  hr.  =  1  day  (da.). 

7  da.  =1  week  (wk.). 

365  da.  =  1  common  year  (yr.) 

366  da.  =  1  leap  year. 

The  year  is  divided  into  12  months,  called  Calendar 
Months,  which  contain  an  unequal  number  of  days, 
namely:  January  31,  February  28,  March  31,  April 
30,  May  31,  June  30,  July  31,  August  31,  September  30, 
October  31,  November  30,  and  December  31. 

Every  fourth  year  contains  366  days,  and  is  called  Leap 
Year,  and  in  these  years  February  has  29  days.  It  is  a 
Leap  Year  when  the  number  of  the  year  is  exactly  di- 
visible by  4 ;  thus,  1896  will  be  a  Leap  Year. 

The  Solar  Year  contains  365  da.  5hr.  48  min.  46  sec,  very 
nearly.  Now  it  would  clearly  be  very  inconvenient  to  reckon  by 
years  which  did  not  contain  an  exact  number  of  days ;  hence,  as 
the  Solar  Year  contains  very  nearly  365^  days,  we  have  3  years 
(called  Civil  Years)  of  365  days  each,  and  then  one  year  of  366  days. 
The  Solar  Year  is,  however,  somewhat  less  than  365^  days,  and  the 
necessary  correction  is  made  by  omitting  three  Leap  Years  in  every 
400  years,  the  years  which  are  not  counted  as  Leap  Years  (although 
divisible  by  4)  are  the  years  which  end  the  Centuries,  and  are  such 
that  the  number  of  the  Century  is  not  divisible  by  4.  Thus,  1800 
was  not  a  Leap  Year,  and  1900  will  not  be  a  Leap  Year ;  the  year 
2000  will,  however,  be  a  Leap  Year. 

158.*   Reduction  of  Compound  Quantities. 

The  method  by  which  a  compound  quantity  can  be 
expressed  as  a  simple  quantity  will  be  seen  from  the 
following  example. 

*  The  methods  of  reductions  of  compound  quantities,  also  ad- 
dition, etc.,  will  be  illustrated  by  the  use  of  the  above  table  because 
the  different  units  are  familiar  to  all. 


Arts.  158-160.]       COMPOUND   ADDITION. 


141 


Ex.  Reduce  7  da.  3  hr.   12  min.   26  sec.  to  seconds. 

7  da.  3  hr.  12  min.  26  sec. 

24 

168 

3 

171  hr. 

60 


10260 

12 

10272  min. 
60 
616320 
26 
616346  sec. 


7  da.  =       168  hr. 

Adding  the  3  hr.,  7  da.  3  hr.  =        171  hr. 

171  hr.  =    10260  mm. 

Adding  12  min.,    171  hr.  12  min.      =a    10272  min. 

10272  min.  =  616320  sec. 

Adding  26  sec,     10272  min.  26  sec.  =  616346  sec. 


6t0)1467t8sec. 
6v0)24t4  min.  38  sec. 

4hr.  4  min.  38  sec. 


159.  To  reduce  a  Simple  Quantity  to  a  Compound  Quantity. 

Ex.  Reduce  14678  sec.  to  hr.,  min.,  and  sec. 

Since  60  sec.  make  lmin.,  if  we 

divide  the  number  of  sec.    by   60, 

we  shall  obtain  the  number  of  min. 

equivalent    to    14678  sec,    i.e.,    244 

min. ,  but  shall  have  38  sec.  over.     We  then  divide  the  number  of 

min.  by  60  and  obtain  the  number  of  hours  with  4  min.  over. 

160.  Addition,  Subtraction,  Multiplication,  and  Division 
of  Compound  Quantities. 

It  will  be  seen  that  no  new  principle  is  involved. 
Care,  however,  must  always  be  taken  in  regard  to  the 
number  of  units  of  one  denomination  required  to  make 
one  unit  of  the  next  higher. 

(a)    Compound  Addition  [see  Art.  142]. 

Ex.  Find  the  sum  of  14  da.  41  min.  11  sec,  121  da.  18  hr.  16  min. 
29  sec,  201  da.  13  hr.  4  sec,  and  11  hr.  23  min.  30  sec. 


da. 


hr. 


14        0  41  11 

121       18  16  29 

201       13  0  4 

11  23  30 

337       19  21  14 


142  NON-DECIMAL   MEASURES.  [Chap.  VI. 

Here  the  sum  of  the  seconds  equals  74  =  1  min.  14  sec. ;  write 
the  14  and  carry  the  1.  The  number  of  min.  =  81  =  1  hr.  21  min. ; 
write  the  21  and  carry  the  1.  The  number  of  hr.  equals  43  =  1  da. 
19  hr.  ;  write  the  19  and  carry  the  1.    The  number  of  days  =  337. 

(6)    Compound  Subtraction. 

Ex.  From  16  da.  12  min.  and  50  sec.  subtract  4  da.  12  hr.  13  min. 
and  54  sec. 


da. 

hr. 

min. 

sec. 

16 

0 

12 

50 

4 

12 

13 

54 

11       11       58      56 

Here  54  cannot  be  subtracted  from  50  ;  therefore  we  take  1  min. 
from  the  12  min.,  change  it  to  sec,  and  we  have  with  the  50  sec. 
110  sec.  in  all  ;  subtract  54  sec.  from  110  sec,  and  we  have  56  sec 
remainder.  Now  13  from  11  we  cannot  take,  therefore  we  take 
1  hr.  from  the  next  column  and  proceed  as  before. 

(c)    Compound  Multiplication. 

Case  I.   When  the  multiplier  is  not  greater  than  12. 

Ex.   Multiply  9  da.  10  hr.  31  min.  14  sec.  by  7. 


da. 

hr. 

min. 

sec. 

9 

10 

31 

14 

7 

66  1  38  38 
Here  14  sec.  x  7  =  98  sec.  =  1  min.  38  sec. ;  write  the  38  and 
carry  1.  31  min.  x  7  =  217  min. ;  217  min.  +  1  min.  =  218  min.  = 
3  hr.  38  min.  ;  write  the  38  and  carry  the  8.  10  hr.  x  7  =  70  hr. ; 
70  hr.  +  3  hr.  =  73  hr.  =  3  da.  1  hr. ;  write  the  1  and  carry  the  3. 
Finally,  9  da.  x  7  =  63  da. ;  63  da.  -f  3  da.  =  66  da.  Ans.  =  66  da. 
1  hr.  38  min.  38  sec. 

Case  II.   When  the  multiplier  can  be  seen  to  be  the 
product  of  factors  each  not  greater  than  12. 
Ex.   Multiply  9  da.  10  hr.  31  min.  14  sec.  by  35. 


9 

10 

31 

14 

7 

66 

1 

38 

38 
5 

330        8      13      10 


Art.  160.]  COMPOUND   DIVISION.  143 

Case  III.  When  the  multiplier  cannot  be  seen  to  be 
the  product  of  factors  each  not  greater  than  12. 

The  following  example  will  explain  the  method  to  be  adopted, 
which  will  he  seen  to  differ  very  little  from  the  method  adopted 
in  the  multiplication  of  simple  quantities,  the  only  apparent  dif- 
ference arising  from  the  fact  that  we  cannot  at  once  write  down 
the  result  of  multiplying  by  10,  100,  etc. 

Ex.   Multiply  9  da.  10  hr.  31  min.  14  sec.  by  257. 

da.       hr.      min.       sec. 


9 

10 

31 

14 
10 

94 

9 

12 

20  = 
10 

multiplicand 

x    10 

943 

20 

3 

20  = 

2 

u 

x  100 

1887 
2d  line  x  5     462 
1st  "    X  7       66 

16 

22 
1 

6 
1 

38 

40  = 
40  = 

38  = 

M 

X2001 
x    50  [ 
x      7j 

2416      15      46      58  =  "  x  257 

(d)   Compound    Division. 

In  division  there  are  two  cases  to  consider,  according 
as  the  divisor  is  an  abstract  number  or  a  concrete 
quantity  of  the  same  kind  as  the  dividend  [Art.  59]. 

Case  I.  To  divide  a  compound  quantity  by  an  abstract 
number. 

Ex.  1.    Divide  22  da.  1  hr.  13  min.  1  sec.  by  6. 

da.  hr.       min.       sec. 

6)22  1  13  1 
3  16  12  10^ 
Here,  dividing  22  da.  by  6,  we  have  3  da.  with  an  undivided  re- 
mainder of  4  da.,  which  must  be  reduced  to  hr. ;  then  we  have  97 
hr.  in  all  to  be  divided  by  6  ;  the  quotient  equals  16  hr.  with  1  hr. 
over.  One  hour  and  13  min.  =  73  min. ;  73  min.  -^  6  =  12  min.  with 
1  min.  over.    Finally,  1  min.  =  60  sec. ;  61  sec.  +  6  =  10^  sec. 

Ex.  2.  Divide  9  wk.  6  da.  21  hr.  13  sec.  by  33. 

wk.      da.       hr.         min.       sec. 

3)9      6      21        0       13 
11)3      2        7        0        41 
2        2      49        5|f 


144  NON-DECIMAL  MEASURES.  [Chap.  VI 

Case  II.   When  the  divisor  is  a  concrete  quantity  of 
the  same  nature  as  the  dividend. 

Ex.   Divide  37  da.  20  hr.  6  min.  48  sec.  by  12  da.  14  hr.  42  min. 
16  sec.     [Compare  Art.  50.] 

37  da.  20  hr.  6  min.  48  sec.  =  3269208  sec. 
12  da.  14  hr.  42  min.  16  sec.  =  1089736  sec. 
3269208  sec.  -=- 1089736  sec.    =  3  (an  abstract  number). 

(e)   To  multiply  or  divide  by  a  fraction. 
[Arts.  121  and  123.] 

Ex.  1.   Multiply  14  da.  2  hr.  12  sec.  by  f . 


da. 

hr. 

min. 

sec. 

14 

2 

0 

12 
5 

)70 

10 

1 

0 

10        1      25        61  f- 
Ex.  2.    Divide  14  da.  2  hr.  12  sec.  by  %. 


da. 

hr. 

min. 

sec. 

►)14 

2 

0 

12 

2 

19 

36 

f 

19       17       12        16f 

Both  operations  are  understood  because  the  nature  of  a  fraction 
has  been  explained. 

EXAMPLES  LIII. 
Written  Exercises. 

1.  Add  17  da.  14  hr.  22  min.  12  sec,  13  da.  11  hr.  24 
min.  18  sec,  and  15  da.  33  min.  40  sec. 

2.  From   6  da.  12  sec.  subtract  2  da.  4  hr.  12  min.  59 
sec. 

3.  Multiply  7  da.  12  hr.  14  min.  25  sec.  by  5. 

4.  Multiply  7  da.  12  hr.  14  min.  25  sec.  by  18. 

5.  Multiply  7  da.  12  hr.  14  min.  25  sec.  by  347. 


Art.  161'.]  AVOIRDUPOIS   WEIGHT.  145 

6.  A  steamer  makes  a  trip  of  800  miles  in  4  da.  8  hr. 
12  min.  20  sec. ;  how  many  such  trips  could  she  make  in 
18  da.  10  hr.  52  min.  25  sec.  ? 

7.  Multiply  8  da.  5hr.  8  min.  48  sec.  by  f . 


Measures  of  Weight. 

The  units  of  measure  for  all  weights  are  derived  from 
the  weight  of  a  kernel  of  wheat  taken  from  the  middle  of 
a  ripe  ear. 

The  name  of  such  a  weight  is  one  Grain  (gr.). 

161.   Table  of  Avoirdupois  Weight. 

The  unit  is  a  pound  consisting  of  7000  grains. 

16  drams  (dr.)     =  1  ounce  (oz.). 
16  oz.  =  1  pound  (lb.). 

100  lb.  =  1  hundred-weight  (cwt.). 

20  cwt.  =  1  ton  (t.). 

112  lb.  =  1  long  hundred-weight. 
2240  lb.  =  1     "    ton  (l.t.). 

Avoirdupois  weight  is  used  in  weighing  all  ordinary  substances. 

The  long  ton  is  used  in  the  Custom  House,  and  in  certain  whole- 
sale transactions. 

The  English  Standard  unit  of  weight  is  the  Imperial  Pound 
(Avoirdupois),  and  is  the  weight  of  a  certain  piece  of  platinum 
kept  in  the  Exchequer  Office. 


English  i 


EXAMPLES  LIV. 
"Written  Exercises. 

1.  Keduce  2  cwt.  151b.  12  oz.  to  dr. 

2.  How  many  gr.  in  an  oz.  avoirdupois  ? 

3.  Add  4  cwt.  721b.  14  oz.  11  dr.,  341b.  12  oz.    2  dr., 
8  cwt.  14  dr.,  14  cwt.  56  lb.  3  oz.,  and  8  lb.  2  oz.  6  dr. 


146  NON-DECIMAL   MEASURES.  [Chap.  VI. 

4.  Eeduce  1687649  dr.  to  units  of  higher  denomina- 
tions. 

5 .  Keduce  16000  oz.  to  long  tons,  etc. 

6.  One  boy  weighs  125  lb.  10.5  oz. ;  how  many  such 
boys  together  weigh  22  cwt.  61  lb.  13  oz.  ? 

162.   Table  of  Troy  Weight. 

The  unit  is  a  pound  consisting  of  5760  grains. 

24  grains  (gr.)  =  1  pennyweight  (pwt.  or  dwt.). 
20  pwt.  =  1  ounce  (oz.). 

12  oz.  =  1  pound  (lb.). 

Troy  weight  is  used  in  weighing  gold  and  silver. 
Diamonds  and  other  jewels  are  spoken  of  as  weighing  so  many 
carats.    The  carat  is  a  little  more  than  3£  grains. 

The  United  States  Standard  unit  of  Weight  is  the  Troy 
Pound  (same  as  the  English  Pound  Troy),  and  is  the 
weight  of  a  certain  piece  of  brass  in  the  custody  of  the 
Director  of  the  U.  S.  Mint. 


EXAMPLES  LV. 
Oral  Exercises. 

1.  How  many  gr.  in  3  pwt.  ?     In  1\  dwt.  ? 

2.  How  many  gr.  in  2  oz.  ? 

3.  How  many  oz.  in  SJ  lb.  ?     In  51  lb.  ? 

4.  How  many  oz.  in  70  pwt.  ?     In  45  dwt.  ? 

5.  How  many  oz.  in  480  gr.  ? 

6.  How  many  lb.  in  78  oz.  ?  In  43  oz.  ?    In  400  pwt.  ? 

Written  Exercises. 

7.  Eeduce  7563  dwt.  to  lb.,  etc. 

8.  Reduce  6  lb.  14  gr.  to  gr. 


Arts.  162, 163.]     APOTHECARIES'    WEIGHT.  147 

9.   How  many  bronze  cents  weigh.  1  lb.,  the  1  ct.  piece 
weighing  48  gr.  ? 

10.  One  hundred  gold  dollar  pieces  weigh  5  oz.  7  pwt. 
12  gr. ;  what  is  the  weight  of  one  piece  ? 

163.   Table  of  Apothecaries'  Weight. 

The  unit  is  a  pound  consisting  of  5760  grains. 

gr.  20  =  1  scruple  (3). 
33  =  1  dram  (3). 
38  =  1  ounce  (5). 
5 12  =  1  pound  (lb.). 

Apothecaries'  weight  is  used  by  physicians  when  writing  pre- 
scriptions and  by  druggists  when  selling  drugs  in  small  quantities. 
Avoirdupois  weight  is  used  by  them  when  dealing  in  large  quantities. 

The  symbols  are  always  written  at  the  left  of  the  figures. 


EXAMPLES   LVI. 
Oral  Exercises. 

1.  How  many  gr.  in  3  3  ?     In  %  1  ?     In3l£? 

2.  How  many  %  in  lb.  2J  ?     In  lb.  3f  ? 

3.  How  many  3  in  gr.  480  ?     In  $  48  ? 

4.  How  many  $  in  5  2  ?     In  gr.  60  ?     In  lb.  1  ? 

5.  How  many  lb.  in  5  42?     In396?    InS103l6? 

Written  Exercises. 

6.  Eeduce  $  7563  to  lb. 

7.  Eeduce  lb.  4  52  to  gr. 

8.  Multiply  lb.6   %1    3l3  2   gr.  15  by  16. 

9.  Divide  ft).  12  S3   3  7  3  2   gr.4by4. 


148  NON-DECIMAL   MEASURES.  [Chap.  VI. 

Measures  of  Length,  Surface,  and  Volume. 

164.   Table  of  Linear  Measures. 

The  English  Standard  unit  of  Length  is  the  Imperial 
Yard  fixed  by  Act  of  Parliament  to  be  the  distance  between 
two  marks  on  a  bar  of  metal  kept  in  the  Exchequer  Office. 

The  U.  S.  Standard  unit  of  Length  is  the  same  as  that 
of  England. 

12  inches  (in.)  =  l  foot  (ft). 

3  ft.  =1  yard  (yd.). 

5*  yd.)  =lrod(rd.). 

16}  ft.  )  V     ' 

320  rd.  =lmile  (mi.). 

For  Land  Surveying. 
7.92  inches  =  1  link  (li.). 

100  li.  |  =1  chain  (ch.). 

4  rd.  ) 

80  ch.  =  1  mi. 

1  mi.  =  1760  yd.  =  5280  ft.=  l  statute  mile. 

Kods  are  sometimes  called  poles  and  perches.  A  furlong  (fur.) 
=  40  rods  =  \  mi.     Civil  engineers  use  a  chain  100  feet  in  length. 

EXAMPLES    LVII. 
Oral  Exercises. 

1.  Express  4  yd.  as  in. ;  7  ft.  as  yd. 

2.  Reduce  18  ft.  to  rd.,  ft.,  and  in. 

3 .  How  many  in.  in  2  yd.  1  ft.  ? 

4.  How  many  ft.  in  a  surveyor's  chain? 

5 .  How  many  li.  in  1  rd.  ? 

6.  How  many  yd.  in  7  rd.  ? 

7.  Practise  frequently  the  drawing  (freehand)  of  a 
straight  line  1  ft.  long. 


Arts.  164,  165.]         SQUARE  MEASURES.  149 

"Written  Exercises. 

8.  Reduce  40  rd.  6  ft.  7  in.  to  in. 

9.  Reduce  1  mi.  to  ft. 
10.    803  in.  to  rd.  and  in. 

165.   Table   of   Square   Measures. 

The  unit  is  any  square,  usually  a  square  which  is  1  ft. 
long. 

144  square  inches  (sq.  in.)  =  l  square  foot  (sq.  ft.). 
9  sq.  ft.  =  1  square  yard  (sq.  yd.) 

m\TS]  =1  square  rod  (sq.rd.) 

160  sq.  rd.  =  1  acre  (A.). 

640  A.  =1  square  mile  (sq.  mi.). 

1A.  =  160  sq.  rd.  =  4840  sq.  yd.  =  43560  sq.  ft. 

For  Land  Surveying. 

16  sq.  rd.  =  1  square  chain  (sq.  ch.). 
10  sq.  ch.  =  1  A. 

Square  measure  is  used  for  measuring  land,  flooring,  and  in  fact, 
everything  in  which  length  and  breadth  have  to  be  taken  into 
account. 

EXAMPLES    LVIII. 
"Written  Exercises. 

1.  Draw  on  the  board  (freehand)  a  figure  representing 
a  square  foot,  marking  it  accurately  into  square  inches. 

2.  How  many  sq.  in.  in  i  a  sq.ft.?  How  many  in 
the  square  of  i  a  ft.  ? 

3.  Having  in  mind  a  square  1  yd.  long,  how  many 
sq.  ft.  in  such  a  square  ? 

4.  Find  by  a  figure  the  number  of  sq.  yd.  in  1  sq.  rd. 


150  NON-DECIMAL  MEASURES.  [Chap.  VI. 

5.  Find   by  multiplication  the  number  of  sq.  yd.  in 
1  sq.  rd. 

6.  How  many  sq.  in.   in   a  square  2   ft.  long  ?     In  a 
square  3 in.  long? 

7.  Kepresent  3  A.  4  sq.  rd.  50  sq.  ft.  as  sq.  ft. 

8.  Divide  58  A.  84  sq.  rd.  3  sq.  yd.  4  sq.  ft.  by  8. 

166.   Table  of  Cubic  Measures. 

The  unit  is  any  cube,  generally  a  cube  1  in.  long,  or  a 
cube  1  ft.  long. 

1728  cubic  inches  (cu.  in.)  =  1  cubic  foot  (cu.  ft.). 
27  cu.  ft.  =  1  cubic  yard  (cu.  yd.). 

For  Measuring  Wood. 

16  cu.  ft.  =  1  cord  foot  (cd.  ft,). 

8  cd.  ft.  =  1  cord  (cd.). 

Cubic  measure  is  used  for  measuring  solid  bodies  in  which 
length,  breadth,  and  thickness  have  to  be  taken  into  account. 


EXAMPLES    LIX. 
Written  Exercises. 

1 .  Make  two  cubes  similar  to  the  one  in  Ex.  2,  p.  136, 
one  cube  an  in.  long,  the  other  4  in.  long. 

2.  Mark  the  sides  of  the  large  cube  into  sq.  in.,  and 
calculate  how  many  cubes  equal  to  the  small  cube  might 
be  cut  from  a  block  equal  to  the  large  cube. 

3.  Reduce  2  cu.  yd.  1201  cu.  in.  to  cu.  in. 

4.  How  many  cu.  in.  in  a  cube  2  in.  long  ?  5  in.  long  ? 

5.  How  many  cords  of  wood  in  a  pile  containing  1541 
cu.  ft.  ?     State  answer  to  two  decimal  places. 

6.  Multiply  18cu.yd.  9cu.ft.  1063  cu.  in.  by  4. 


Arts.  166-169.]         LIQUID  — DRY  — FLUID.  151 

167.  Table  of  Liquid  Measures. 

The  unit  is  a  Gallon  of  231  cu.  in.  (the  old  English 
wine  gallon). 

4  gills  (gi.)=  1  pint  (pt.). 
2  pt.  =1  quart  (qt.). 

4  qt.  =1  gallon  (gal.). 

311  gal.  =  1  barrel  (bbl.). 

2  bbl.  =  1  hogshead  (hhd.). 

A  gallon  of  water  weighs  8.33  lb. 

The  quart  is  a  volume  of  57  £  cu.  in. 

The  English  Imperial  Gallon  contains  277.274  cu.  in. 

168.  Table  of  Dry  Measures. 

The  unit  is  a  Bushel  of  2150.42  cu.  in.  (the  old  English 
Winchester  bushel). 

2  pints  (pt.)=  J  quart  (qt.). 
8  qt.  =1  peck  (pk.). 

4  pk.  =  1  bushel  (bu.). 

The  quart  is  a  volume  of  67 £  cu.  in. 

The  English  Imperial  bushel  is  8  Imperial  gallons  =  2218.192 
cu.  in. 

169.  Table  of  Apothecaries'  Fluid  Measures. 

60  minims,  or  drops  (i*L)=l  fluid  dram  (f3). 
f38  =1  fluid  ounce  (fS). 

fS16  =lpint(0). 

08  =1  gallon  (Cong.). 

EXAMPLES  LX. 
Oral  Exercises. 

1.  How  many  pt.  in  5  qt.?     In  3  gal.? 

2.  How  many  pt.  in  3  pk.?     In  1  bu.? 

3.  Is  the  pt.  in  Ex.  1  equal  to  the  pt.  in  Ex.  2  ? 

4.  Eeduce  3  qt.  1  pt.  to  gi.  j  1  bu.  to  qt. 


152  NON-DECIMAL   MEASURES.  [Chap.  VI. 


5.  How  many  «t  in  f§l? 

6.  How  many  f§  in  ^960? 


Written  Exercises. 

7.  Eednce  Cong.l  to  K. 

8.  Divide  14bu.  3  pk.  5  qt.  1  pt.  by  5. 

9.  Divide  0  7  fglO  f36  "l59§  by  f. 

10.  Keduce  750  dry  qt.  to  liquid  qt. 

11.  How  many  bushels  of  potatoes  in  a  bin  4  ft.  x  3  ft. 
X  2  ft.  ? 

12.  How  many  piles  of  3  bu.  each  could  be  made  out 
of  a  pile  containing  8  cu.  ft.  ? 

13.  Reduce  1  hhd.  to  cu.  ft. 

Foreign  Monies. 

170.  Table  of  English  Money. 
The  unit  is  the  Pound  (£). 

4  farthings  (/cw\)  =  l  penny  (&). 
12d  =  1  shilling  (*.). 

20s.  =  1  pound  (£). 

21s.  =1  guinea  (ga.). 

One  farthing  is  written  \d.  ;  two  farthings,  or  one  half -penny, 
is  written  \d. ;  and  three  farthings  is  written  \d.  Thus,  eightpence 
farthing  is  written  S\d. 

The  coins  in  use  in  England  are  as  follows  : 
Gold  coins  :   the  sovereign  (20s.)  and  half-sovereign  (10s.). 
Silver   coins:    crown   (5s.),  half-crown  (2s.  6d.),  florin   (2s.), 
double  florin  (4s.),  shilling,  sixpence,  and  threepence. 
Copper  coins :   penny,  half-penny,  and  farthing. 

171.  German  Money. 

100  pfennigs  (j)/.)  =  l  mark  (m.). 


Arts.  170-174.]  FRENCH    MONEY.  153 

172.  French  Money. 

100  centimes  (c.)  =  1  franc  (/.). 
Note.    See  Art.  262  for  equivalents  in  American  money. 

173.  The  value  of  a  given  fraction  of  a  given  concrete 
quantity  is  found  as  follows  : 

For  example,  to  find  T5f  of  3  lbs.  4  oz.  Troy,  we  must  divide 
3  lbs.  4  oz.  into  18  equal  parts,  and  then  take  5  of  those  parts  ; 
that  is,  we  must  divide  by  18,  and  then  multiply  by  5.  We  may, 
however,  first  multiply  by  5  and  then  divide  by  18.     Thus, 


18 


174.   The  value  of  a  given  decimal  of  a  given  concrete 
quantity  is  found  as  follows  : 

Ex.    Find  .54375  of  1  lb.  Troy  in  lower  denominations. 
.54375 


lb.    oz.  pwt.       gr. 

13_4 
|l     8 

lb. 

3 

oz.    pwt.    gr. 

4 

5 

2    4     10} 

b 

«}S 

16 

8 

8 

4 

11     2      5£ 

11     2     5} 

Here  .54375  of  1  lb.  =  .54375  of  12  oz., 
which  =  6.525  oz.  ; 

.525  of  loz.  =  .525  of  20  pwt., 
which  =  10.5  dwt. ; 

.5  of  ldwt.  =.5  of  24  gr., 
which  =  12  gr. 


12 


20 


dwt.  =10.500 
24 


gr.      =  12.0 


.  \  6  oz.  10  dwt.  12  gr.  =  Ans. 

EXAMPLES    LXI. 
Written  Exercises. 


1.  Find  2|  of  3  da.  12  hr. 

2.  Find  4f  of  3  cwt.  36  lb. 

3.  Add  |  of  2s.  6d,  2£  of  Is.  Sd.,  and  l^  of  6s.  5d 


154  NON-DECIMAL   MEASURES.  [Chap.  VI. 

4.  Find  1.625  of  1  da.  -  .02  of  1  wk. 

5.  Express  4.3125  lb.  Troy  in  lb.,  oz.,  etc. 

6.  Find  .436  of  1  mi. 

7.  By  how  much  does  f  of  1  mi.  exceed  ^  of  310  rd. 
lyd.? 

175.  To  express  one  quantity  as  a  fraction  of  another, 

we  proceed  as  follows : 

Ex.  1.    Express  14s.  6d.  as  a  fraction  of  lbs.  Sd. 

Us.  6d.  =  lUd. 
lbs.  Sd.  =  188c?. 

Now  Id  =  %h  of  188<*- 

...  i74tf.  =  i|4  of  issd. 

Ex.  2.    Express  2£  of  Is.  7\d.  as  a  fraction  of  £1. 

2£  of  Is.  l\d.  =  -U  of  ¥*  =  -4tW- 
£1  =  240<J. 
Now  Id.  =  j£*  of  £1. 

.  •.  Wd.  =  W  x  &  of  £1  =  £  ^\\ 


176.   To  express  one  quantity  as  a  decimal  of  another, 

we  proceed  as  follows : 

The  method  here  is  the  reverse  of  that  in  Art.  174. 

Ex.  Express  Id  oz.  II  dwt.  12  gr.  as  the  decimal  of  lib.  Troy. 
12  gr.  Divide  the  grains  by  24  to  reduce  to  pwt.;   add 


11.6  the  11  pwt.  and  divide   11.5  pwt.  by  20  to  reduce 


10.575        to  oz ;    add  the  10  oz.  and  divide  10.675  oz.  by  12 


.88125    to  reduce  to  lb. 


An  excellent  method  is  to  express  one  quantity  as  a 
fraction  of  the  other  [Art.  175],  and  then  reduce  this 
common  fraction  to  a  decimal  [Art.  130] ;  thus, 


Arts.  175,  176.]  EXAMPLES.  155 

10  oz.  lldwt.  12  gr.  =  5076  gr. 
1  lb.  =  5760  gr. 

Now,  *W  ='88125' 


EXAMPLES  LXII. 
Written  Exercises. 

1.  Express  251b.,  601b.,  121b.  8  oz.,  and  61b.  4oz.  as 
fractions  of  1  cwt. 

2.  What  would  be  the  measure  of  4  yd.  2  ft.  8  in.  if 
1  yd.  1  ft.  7  in.  were  taken  as  the  unit  ? 

3.  Express  1  oz.  6  dwt.  6  gr.  as  a  decimal  of  1  lb.  Troy. 

4.  What  decimal  of  an  acre  is  20  sq.  rd.  5  sq.  ft.  72 
sq.  in.? 

5.  Express  £5.  12s.  6&  as  a  decimal  of  £10. 

6.  Express  2  mo.  7  da.  as  a  decimal  of  1  yr. 

7.  Express  7  mo.  12  da.  as  a  decimal  of  1  yr. 

8.  Express  10  mo.  15  da.  as  a  decimal  of  2  yr. 

EXAMPLES   LXIII. 
Simple  Examples  in  Reduction  for  Written  Work. 

Reduce : 

1 .  1 1.  3  cwt.  10  lb.  to  pounds. 

2.  3t.  12  cwt.  161b.  to  pounds. 

3.  6hr.  12min.  10  sec.  to  seconds. 

4.  12  hr.  5  min.  24  sec.  to  seconds. 

5.  13  yd.  2  ft.  11  in.  to  inches. 

6.  17  yd.  2  ft.  7  in.  to  inches. 

7.  12  mi.  3  fur.  10  rd.  to  rods. 

8.  13  mi.  5  fur.  26  rd.  to  rods. 

9.  8  bu.  3  pk.  4  qt.  to  quarts. 


156  NON-DECIMAL  MEASURES.  [Chap.  VI. 

10.  5  gal.  3  qt.  1  pt.  to  pints. 

11.  5  A.  27  sq.  rd.  to  square  rods. 

12.  17  A.  135  sq.  rd.  to  square  rods. 

13.  13  sq.  yd.  6  sq.  ft.  100  sq.  in.  to  inches. 

14.  8  sq.  yd.  7  sq.  ft.  90  sq.  in.  to  inches. 

15.  61b.  7  oz.  10  dwt.  15  gr.  to  grains. 

16.  18  lb.  9  oz.  15  dwt.  20  gr.  to  grains. 

17.  3  wk.  5  da.  12  hr.  to  hours. 

18.  16  da.  22  hr.  40  min.  35  sec.  to  seconds. 

19.  12 1.  13cwt.  751b.  7  oz.  to  ounces. 

20.  5 1.  17  cwt.  68  lb.  14  oz.  to  ounces. 

21.  2  mi.  3  fur.  80  yd.  2  ft.  to  feet. 

22.  12  mi.  1200  yd.  1ft.  7  in.  to  inches. 
Reduce  to  tons,  cwt.,  etc. : 

23.  14621b.  25.    115971b.  27.    57812  oz. 

24.  135741b.         26.    56214  oz.  28.    81974  dr. 

Reduce  to  acres  and  square  rods : 

29.  315  sq.  rd.  31.    1574  sq.  rd. 

30.  5142sq.rd.  32.  3725  sq.  rd. 

Reduce  to  yards,  feet,  etc. : 

33.  156  in.   34.  342  in.   35.  417  in.   36.  1179  in. 

Reduce  to  lb.,  oz.,  dwt.,  gr. : 

37.  517  dwt.  41.  13407  gr. 

38.  574  dwt.  42.  24709  gr. 

39.  3156  gr.  43.  35937  gr. 

40.  4215  gr.  44.  51940  gr. 

Reduce  to  bushels,  pecks,  etc. : 

45.  156  pt.  46.  1472  pt.  47.  416  qt.   48.  1875  pt. 


Art.  177.]  DIFFICULT  REDUCTIONS.  157 

Keduce  to  square  yards,  etc. : 

49.    1462  sq.  in.       50.    2156sq.in.       51.    3564  sq.  in. 
Keduce  to  days,  hours,  etc. : 

52.   31572  sec.         53.   257672  sec.        54.    7142169  sec. 
Calculate  the  number  of 

55.    Sq.  rd.  in  1  sq.  mi.  56.    Sq.  rd.  in  1  A. 

57.    A.  in  1  sq.  mi. 

58.  Change  3.12  rd.  to  the  decimal  of  a  mi. 

59.  Change  .2  sq.  rd.  to  the  fraction  of  an  A. 

60.  Change  lb.  .00694  to  the  fraction  of  a  3. 

61.  How  many  cd.  of  wood  might  be  packed  into  a 
shed  the  size  of  your  school-room  ? 

177.  The  following  cases  are  somewhat  more  difficult 
than  those  previously  considered  because  one  rod  does 
not  equal  an  exact  number  of  yards. 

Ex.  1.   Beduce  Bird.  3yd.  2  ft.  Win.  to  inches. 

31rd.  3  yd.  2  ft.  11  in. 
5.5 


15  5 

155  Here  we  have  31  rods  to  be  multi- 

173.5  yd.,  the  3  yd.  included,    plied  by  5^  =  5.5.     We  might  have 

§  multiplied  by  JUk 

522.5  ft.,     "2  ft. 
12 


6281.0  in.,     "  11  in. 

Caution.     In  adding  the  yards  or  feet  of  the  example  while 
multiplying,  care  must  be  used  in  regard  to  the  decimal  point. 

Ex.  2.   Beduce  1885  sq.  rd.  16  sq.  yd.  6  sq.ft.  to  sq.ft. 

1885  sq.  rd. 
30.25 
57037.25  sq.  yd.,  including  the  16sq.yd. 
9 


513341.25  sq. ft.,  "  "      6  sq.ft. 


158  NON-DECIMAL  MEASURES.  [Chap.  VI. 

Ex.  3.  Reduce  6281  in.  to  units  of  higher  denominations. 

First  Method. 

12)6281  in.  5.5,)174.0%(31 

3)523  ft.  5  in.  165 

5.5)174  yd.  1ft.  ?0 

31  rd.  3.6  yd. 


55 

35     [See  Art.  68.] 


rd.  yd.       ft.  in. 

Integral  part  of  answer               =31  3      1        5 

Decimal    "    "       M      =.5  yd.  =  1        6 

Sum  =  31  3      2  11    Ans. 

Second  Method. 


12)6281  in. 
3)523  ft. 

5  in. 

174  yd. 
2 
11)348 
31  rd. 

1  ft.                               It  is  shorter  to  multiply  by  T2T 
than  to  divide  by  5.5. 

.  .  .  1  yd.  [Art.  68.] 

rd.        yd.       ft.         in. 

Integral  part                   =  31      0      1        5 

Fractional  part  =  f  yd.  =           3      1        6 

Sum  =  31      3      2      11    Ans. 

Ex.  4.    Beduce  513341.25  sq.ft.  to  sq.  rd.,  sq.  yd.,  and  sq.ft. 

9)513341.25 
30.25)57037      sq.  yd.        8.25  sq.  ft. 
1885      sq.rd.       15.75  sq.  yd. 

sq.  rd.     sq.  yd.    sq.  ft.    sq.  in. 

Integral  part  =  1885      15      8 

_     .      .        .    r  .25  sq.  ft.  =  36 

Decimal  part  i    _-  ,  ~ 

*        I  .75sq.yd.  =  6      108 

Sum  =  1885      16      6  Ans. 

Here  it  is  shorter  and  easier  to  divide  by  30.25  than  to  divide  by 
ifJ-  (i.e.,  to  multiply  by  T|T). 

It  will  be  noticed  that  only  the  integral  part  of  any  dividend  is 
to  be  divided ;  the  decimal  part,  if  any,  is  to  be  regarded  as  a 
decimal  part  of  the  remainder. 


Art.  178.]  MISCELLANEOUS  MEASURES.  159 

178.   In  some  cases    of    Reduction  we  cannot  pass 
directly  from  one  denomination  to  the  other. 
Ex.   How  many  lb.  Troy  are  there  in  144  lb.  Avoir,  f 

Since  1  lb.  Avoir.  =  7000  gr., 

144  lb.  Avoir.  =  7000  gr.  x  144. 

These  grains  are  now  reduced  to  lb.  Troy  in  the  usual  manner. 


Stationery. 


Miscellaneous  Measures.  Numbers. 

3  barleycorns  =  1  in.  12  units    =  1  dozen. 

4  in.  =1  hand.  12  dozen  =  1  gross. 

40  rd.  =1  furlong.  12  gross    =  1  great  gross. 

i   ~~~~.~i.t  ->  20  units    =  1  score. 

1  geographi-)       lknot# 

cal  mi.  =6080  ft.  J 

3  knots  =  1  league. 

6  ft.  =1  fathom.  24  sheets     =  1  quire. 

20  quires     =  1  ream. 

1  cu.  ft.  of  pure  water  weighs  2  reams     =  1  bundle. 

1000  oz.  =  62 1  lb.  5  bundles  =  1  bale. 

EXAMPLES  LXIV. 
Reduce  :  Written  Exercises. 

1.  10  rd.  2  yd.  1ft.  to  feet. 

2.  5rd.  3  yd.  2  ft.  to  inches. 

3.  1  mi.  3  fur.  20  rd.  1  yd.  to  yards. 

4.  6  mi.  5  fur.  30  rd.  3  yd.  to  yards. 

5.  18  mi.  11  rd.  3  yd.  1ft.  6  in.  to  inches. 

6.  27  mi.  273  rd.  2  yd.  2  ft.  7  in.  to  inches. 

7.  6  mi.  52  yd.  to  yards. 

8.  18  mi.  5  rd.  160  yd.  2  ft.  11  in.  to  inches. 

9.  3  A.  16  sq.  rd.  to  square  yards. 
10.  15  A.  24  sq.  rd.  to  square  yards. 

*The  knot  recognized  by  the  U.  S.  Coast  and  Geodetic  Survey 
equals  6080.20  ft. 


160  NON-DECIMAL   MEASURES.  [Chap.  VI. 

11.  3  A.  85  sq.  rd.  16  sq.  yd.  6  sq.  ft.  to  square  inches. 

12.  16  sq.  rd.  18  sq.  yd.   5  sq.  ft.   100  sq.  in.  to   square 


inches. 

Reduce  to  miles,  etc. : 

13.  6974  yards. 

16. 

6315  feet. 

14.  21571  yards. 

17. 

51621  inches. 

15.  15737  yards. 

18. 

158743  inches. 

Reduce  to  acres,  sq.  rd., 

etc. 

19.  20812  sq.  yd. 

21. 

5172400  sq.  in. 

20.  38599  sq.  yd. 

22. 

8156179  sq.  in. 

Reduce : 

23.  36  lb.  Avoir,  to  lb.  Troy. 

24.  720  lb.  Avoir,  to  lb.  Troy. 

25.  1  cwt.  Avoir,  to  Troy  weight. 

26.  11  lb.  8  oz.  Avoir,  to  Troy. 

27.  350  oz.  Troy  to  oz.  Avoir. 

28.  4  lb.  3  oz.  20  gr.  to  lb.  and  oz.  Avoir. 

29.  lcwt.  91b.  to  lb.,  3,  etc. 

30.  lb.  9   56  3  6  3  2  gr.5  to  lb.,  etc.,  Avoir. 

EXAMPLES  LXV. 
Written  Exercises. 

Add: 

da.       hr.  mini.  da.       hr.       min.       sec. 

1.       5    17  42  3.       5     17     27    45 

3    11  53  6    11    39    56 

7     19  37  17    21     49    40 

11      7  21  6     11     11     31 


2. 


hr.  min.   sec. 

cwt.   lb. 

oz. 

1  41  15 

4   5  16 

10 

6  17  39 

3  39 

6 

7  35  42 

7  -47 

14 

5  16  13 

1  25 

9 

Art.  178.]  EXAMPLES.                                       161 

lb.        oz.  dr.  lb.  oz.  dwt.  gr. 

5.       5     12  8  12.       5  11  16  18 

4    13  12  2  9  11  13 

7      9  15  7  10  15  21 

3    11  14  3  7  9  16 


t.  cwt.  lb.  oz.  yd.  ft.  in. 

5  15  17  3  13.       5  2  9 

1  12  67  12  11  1  0 

15  17  20  11  13  2  7 

3  9  21  7  6  0  11 


lb.  oz.  dr.  yd.  ft.  in. 

7.       5  9  13  14.     16  1  7 

7  14  12  9  2  10 

18  6  9  20  0  8 

3  11  11  11  2  11 


t.  cwt.  lb.  oz.  yd.  ft.  in. 

8.     16  17  19  14  15.     15  0  9 

119  16  47  0  3  2  7 

72  12  37  13  18  1  11 

65  15  24  8  8  0  9 


mi.    rd.   yd. 

9.       6    24    10  16.       6    100    2 


cwt. 

lb. 

oz. 

6 

24 

10 

17 

78 

12 

14 

7 

14 

11 

41 

2 

lb. 

oz. 

dwt. 

6 

4 

19 

13 

9 

7 

2 

11 

17 

7 

10 

13 

oz. 

dwt. 

gr. 

1 

17 

23 

2 

8 

11 

5 

15 

7 

7 

4 

21 

3    140    4 

18      97    3 

2      15    2 


mi.    rd.  yd.  ft.  in. 

10.                                               17.       1     190  2  1  4 

3  3  0  11 

2  84  4  2  7 

3  180  3  1  9 


mi.    rd.   yd.   ft.    in. 

1.  17    23  18.       5    300    2     2      1 


15     3     1       9 

1  187     4    2    11 

2  74    5    0      9 


162  NON-DECIMAL  MEASURES.  [Chap.  VI. 


A. 

sq.  yd. 

bu. 

Pk. 

qt. 

pt. 

19. 

5 

12 

23. 

3 

2 

5 

1 

17 

25 

1 

3 

3 

0 

3 

18 

10 

0 

6 

1 

4 

30 

2 

3 

4 

1 

A. 

sq.  yd. 

m. 

5 

3 

9 

gr. 

20. 

1 

27 

24. 

4 

10 

6 

2 

5 

16 

19 

3 

8 

5 

2 

15 

8 

22 

1 

0 

1 

1 

6 

19 

7 

2 

0 

7 

0 

19 

gal. 

qt.   pt. 

Cong. 

0. 

*3 

f3 

21. 

5 

2  1 

25. 

1 

6 

12 

4 

6 

3  1 

2 

5 

13 

7 

4 

1  0 

1 

2 

3 

5 

1 

gal. 

2  1 

qt.   pt. 

gi- 

2 

1 

5 

3 

lb. 

5 

3 

3 

22. 

18 

3  1 

2 

26. 

1 

11 

7 

2 

4 

1  0 

3 

2 

9 

1 

1 

6 

2  1 

1 

3 

6 

6 

0 

1 

1  .1 

1 

cu.  yd. 

cu.ft. 

5 

cu.  in. 

4 

5 

1 

27. 

5.2 

22.1 

16.4 

1.3 

19.2 

126.9 

3.3 

3. 

14.3 

5.4 

8.2 

9.2 

Answer  in  exact  units. 

EXAMPLES  LXVI. 
"Written  Exercises. 

Subtract : 

1.  5  da.  16  hr.  22  min.  from  11  da.  18  hr.  lOmin. 

2.  15  da.  17  hr.  13  min.  42  sec.  from  31  da.   9hr.  11 
min.  40  sec. 

3.  5  cwt.  73  lb.  11  oz.  from  7  cwt.  11  lb.  9  oz. 


Art.  178.]  EXAMPLES.  163 

4.  61b.  10  oz.  11  dr.  from  161b.  9  oz.  5  dr. 

5.  7t.  13cwt.  151b.   12  oz.  from  10 1.  11  cwt.  10  oz. 

6.  3  lb.  4  oz.  10  dwt.  from  9  lb.  1  oz.  5  dwt. 
Find: 

7     731b.4oz.l0pwt._2(51b  10oz  18pwt  1Q5gr) 
o 

8.  10  yd. -5  yd.  1ft.  10  in. 

9.  29  yd.  1ft.  4  in. -17  yd.  2  ft.  11  in. 

10.  17  mi.  lfur.  150  yd. -6  mi.  3  fur.  164  yd. 

11.  From  lb.  4  56  gr.17  subtract  lb.  2  %1  3  3   gr.  15. 

12.  From   18  sq.  yd.    3  sq.  ft.    17  sq.  in.    take  6  sq.yd. 
7  sq.ft.  lOOsq.in. 

13.  From  215  sq.  yd.  3  sq.  ft.  84  sq.  in.  take  118  sq.yd. 
6  sq.  ft.  112  sq.  in. 

14.  From  25  A.  take  15  A.  120  sq.  rd.  10  sq.  yd. 

15.  From  23  A.  40  sq.  rd.  10  sq.yd.  take  6  A.  125 
sq.  rd.  25  sq.  yd. 

16.  Find  6  cu.  yd.  24  cu.  ft.  1200  cu.  in.  -  3  cu.  yd.  25 


cu.  ft.  8  cu.  in. 

bu.       pk.          qt.        pt.        gi. 

17.    7     1.2        0     1     3.6 
3    2.4    1.3    0    2.4 

18. 

gal. 

10 
5 

qt.      pt. 

1 

2     1 

Answer  in  exact  units. 

0. 
19.    7 

*3 
10 

f3 
5 

50 

3 

14 

6 

51 

EXAMPLES  LXVII. 
Written  Exercises. 


Multiply : 

1.  5  hr.  10  min.  33  sec,  (i)  by  5,  (ii)  by  7,  (iii)  by  9. 

2.  5  cwt.  39  lb.,  (i)  by  7,  (ii)  by  8,  (iii)  by  9. 


164  NON-DECIMAL   MEASURES.  [Chap.  VI. 

3.  6t.   17cwt.  641b.  6oz.  5  dr.,  (i)  by  4,  (ii)  by  6, 
(iii)  by  9. 

4.  8  lb.  10  oz.  15  dwt.  20  gr.,  (i)  by  5,  (ii)  by  7,  (iii) 
by  12. 

5.  tt>6   34.1   32.3   31.2  gr.ll  by  5. 

6.  10  yd.  1  ft.  7  in.,  (i)  by  8,  (ii)  by  11,  (iii)  by  12. 

7.  8  mi.  215  yd.,  (i)  by  5,  (ii)  by  8,  (iii)  by  12. 

8.  1  mi.  20  rd.  4  yd.,  (i)  by  7,  (ii)  by  56. 

9.  15  sq.  yd.  7  sq.  ft.  100  sq.  in.,  (i)  by  6,  (ii)  by  11. 

10.  4  en.  ft.  163  cu.  in.,  (i)  by  8,  (ii)  by  11. 

11.  3  bu.  2  pk.,  (i)  by  5,  (ii)  by  11. 

12.  3  gal.  2  qt.  1  pt.,  (i)  by  5,  (ii)  by  7. 

13.  3  da.  17  hr.  10  min.  15  sec,  (i)  by  35,  (ii)  by  45. 

14.  15 1.  12  cwt.  16  lb.,  (i)  by  42,  (ii)  by  72. 

15.  8  lb.  11  oz.  15  dwt.  18  gr.,  (i)  by  49,  (ii)  by  84. 

16.  3  yd.  2  ft.  10  in.,  (i)  by  44,  (ii)  by  132. 

17.  3  yd.  1ft.  7  in.  by  350. 

18.  5bn.  2pk.  by  420. 

19.  12  da.  13  hr.  14  min.  12  sec.  by  65. 

20.  5t.  7  cwt.  151b.  by  94. 

21.  31b.  4oz.  12  dwt.  12  gr.  by  124. 

22.  3  cwt.  75  lb.  5  oz.  by  257. 

23.  15sq.yd.  7  sq.ft.  82sq.in.  by  1212. 

24.  6t.  15  cwt.  71b.  3  oz.  by  2341. 

25.  21b.  4oz.  16  dwt.  18  gr.  by  3124. 

26.  1  mi.  2  fur.  15  rd.  4  yd.,  (i)  by  5,  (ii)  by  9. 


Art.  178.]  EXAMPLES.  165 

EXAMPLES  LXVIII. 
Written  Exercises. 

Divide : 

1.  22  da.  lhr.  12min.  by  6. 

2.  37  cwt.  3  lb.  by  7. 

3.  441b.  2oz.  8  dr.  by  8. 

4.  52  lb.  10  oz.  13  dwt.  by  9. 

5.  153  yd.  2  ft.  lin.  by  11. 

6.  95  A.  64sq.rd.  by  12. 

7.  1851b.  8oz.  17  dwt.  by  54. 

8.  123  da.  10  hr.  45  min.  by  50. 

9.  1052  yd.  1ft.  by  132. 

10.  251  A.  133sq.rd.  by  121. 

11.  19 1.  14  cwt.  81b.  3oz.  4  dr.  by  500. 

12.  214 1.  10  cwt.  441b.  by  196. 

13.  12 1.  3  cwt.  9  lb.  by  37. 

14.  309 1.  12  cwt.  141b.  by  47. 

15.  lOt.  6  cwt.  701b.  loz.  by  57. 

16.  37  yd.  2  ft.  3  in.  by  151. 

17.  35 1.  2  cwt.  631b.  2  oz.  by  289. 

18.  2237  bu.  1  pk.  7  qt.  by  253. 

19.  61 1.  1  cwt.  75  lb.  by  2896. 

20.  24  mi.  58  yd.  2  ft.  4  in.  by  1234. 

21.  36  mi.  4  fur.  23  rd.  3  yd.  1ft.  6  in.  by  10. 
•   22.  55  mi.  7  fur.  26  rd.  1yd.  1ft.  by  43. 

23.  298  A.  39  sq.  rd.  18  sq.  yd.  2  sq.  ft.  108  sq.  in.  by  73. 


166  NON-DECIMAL   MEASURES.  [Chap.  VI. 

EXAMPLES    LXIX. 
Written  Exercises. 

1.  Divide  2  tons  5  cwt.  by  9  cwt. 

2.  Divide  6  oz.  10  dwt.  by  13  dwt. 

3.  Divide  3  A.  50  sq.  rd.  by  19  sq.  rd. 

4.  Divide  20  bu.  1  pk.  by  2  bu.  1  pk. 

5.  How  many  pieces  each  3  yd.  1  ft.  long  can  be  cut 
from  a  rope  whose  length  is  180  yd.? 

6.  A  wheel  revolves  once  every  2  m.  15  sec. ;  how 
many  times  does  it  revolve  in  1  hr.  48  m.  ? 

7.  The  circumference  of  a  tricycle  wheel  is  12  feet; 
how  many  times  does  the  wheel  turn  round  in  a  journey 
of  10  miles  ? 

8.  A  field  of  13  A.  80  sq.  rd.  is  divided  into  allot- 
ments, each  containing  1  A.  20  sq.  rd. ;  how  many  allot- 
ments are  there  ? 

9.  A  man's  average  step  is  2  ft.  11  in. ;  how  many 
steps  does  he  take  in  walking  3  J  miles  ? 

10.  How  many  jars,  each  containing  2  gal.  3  qt.  1  pt., 
can  be  filled  out  of  a  cask  containing  46  gal.  ? 

11.  How  many  rails,  each  weighing  4  cwt.  37  lb.,  can 
be  made  out  of  58 1.  19  cwt.  90  lb.  of  iron  ?  What  will 
each  rail  cost  at  3  ct.  a  lb.  ? 

12.  How  many  times  does  2  miles  76  yd.  contain  14 
yd.  1ft.  6  in.  ? 

13.  Each  of  a  certain  number  of  articles  weighs  14  lb. 
1  oz.,  and  the  total  weight  is  3  t.  75  lb. ;  how  many  are 
there  ? 

14.  How  many  times  is  361b.  3oz.  3  dwt.  contained  in 
543  lb.  11  oz.  5  dwt.  ? 


Art.  179.] 


CIRCULAR    MEASURES. 


167 


15.  How  many  bullets,  each,  weighing  2  \  oz.,  can  be 
made  from  a  quantity  of  lead  weighing  7  cwt.  35  lb.  ? 

16.  A  sovereign  weighs  123  grains ;  how  many  can  be 
made  out  of  3  lb.  5  oz.  of  standard  gold  ? 

179.   Table  of  Circular  Measures. 

The  plane  figure  whose  bound- 
ing line  is  a  curve  everywhere 
equally  distant  from  the  centre 
is  called  a  Circle. 

The  bounding  line  of  a  circle 
is  called  its  Circumference. 

Any  part  of  a  circumference 
is  called  an  Arc. 

If  the  circumference  be  di- 
vided into  360  equal  parts,  one  of  these  parts  is  called  an 
arc  of  one  Degree  (1°). 

The  unit  is  an  arc  of  1°. 
60  seconds  (")  =  1  minute  (*). 
60'  =  1  degree  (°). 

360°  =  1  circumference  (C). 


EXAMPLES  LXX. 
Written  Exercises. 


.   Ac 

Id         5° 

21' 

15" 

27° 

41' 

23" 

196° 

12' 

39" 

150° 

2' 

10" 

2.    From        182°     1'    49" 
Subtract    12°  50'    50" 


3.  How  many  seconds  in  90°  ? 

4.  How  many  degrees  in  5678"  ? 

5.  How  many  circumferences  in  1800°  ? 

6.  Eeduce  100000"  to  units  of  higher  denominations. 


168 


NON-DECIMAL  MEASURES. 


[Chap.  VI. 


180.  Longitude  and  Time. 

EXAMPLES  LXX.  —  Continued. 
Oral  Exercises. 

7.    Let  the  figure  represent  a  globe   rotating  on   its 
axis ;  how  many  degrees  does  c  move  towards  the  present 


position  of  g  while  the  globe  is  making  I  of  a  rotation 


•L  of  a  rotation  5 


8.  The  earth  is  a  rotating  globe,  and  a  point,  as  c 
or  r,  moves  once  around  its  circle  in  24  hr. ;  how  long 
does  it  take  c  to  move  to  the  present  position  of  e,  the  arc 
ce  being  30°  ?     To  the  present  position  of  g  ?     Of  d  ? 

9.  How  long  does  it  take  r  to  move  to  the  present 
position  of  s  ? 

10.  How  long  does  it  take  the  arc  ac  to  reach  the 
present  position  of  the  arc  (meridian)  ae  ? 

11.  How  many  degrees  does  the  earth  rotate  in  1  hr.  ? 
In  1  min.  ? 


Arts.  180,  181.]      LONGITUDE   AND   TIME. 


169 


12.  How  many  arc  minutes  does  the  earth  rotate  in  1 
min.  ?     In  1  sec.  ? 

13.  How  many  arc  seconds  does  the  earth  rotate  in 
1  sec.  ? 

Since  15°  rotation  require  1  hr. 

and  15'  rotation  require  1  min. 

and  15"  rotation  require  1  sec, 


we  may  change  time  measure  to  circular  measure  by  mul- 
tiplying hr.,  min.,  and  sec.  by  15 ; 

we  may  change  circular  measure  to  time  measure  by  dividing 
°,  ',  and  "  by  15. 

The  meridian  distance  (the  difference  in  longitude) 
between  two  places  is  measured  in  units  of  circular  meas- 
ure, or  in  units  of  time  measure. 

181.   Difference  in  Longitude,  and  in  Time. 
Longitude  is  reckoned  either  east  or  west  from  the  merid- 
ian passing  through  Greenwich.     It  is  evident  that  if  two 


170  NON-DECIMAL  MEASURES.  [Chap.  VI. 

places  are  either  east  of,  or  west  from,  Greenwich,  the  dif- 
ference in  longitude  is  found  by  subtraction ;  if  one  place  is 
east  and  the  other  west,  the  difference  is  found  by  addition. 

Ex.   Find  difference  in  time  between  Cleveland,  81°  40'  30"  W., 
and  St.  Paul,  93°  4'  55"  W. 

93°    4' 55" 

81°  40'  30" 

15)11°  24'  25" 

45  min.  37  sec.  Ans. 

EXAMPLES  LXXI. 
0  "Written  Exercises. 

Find  the  difference  in  time  between 

1.  Portland  (Me.),  70°  15'  40"  W.,  and  Detroit,  82° 
58' W. 

2.  New  York,  74°  0'  3"  W.,  and  Chicago,  87°  37'  30"  W. 

3.  New  York  and  Washington,  77°  2'  48"  W. 

4.  Berlin,  13°  23'  53"  E.,  and  Paris,  2°  20'  22."5  E. 

5.  Berlin  and  New  York. 

6.  Boston,  71°  3'  30"  W.,  and  San  Francisco,  122°  24' 
15"  W. 

7.  Greenwich  and  Washington. 

8.  What  is  the  longitude  of  St.  Louis,  the  difference 
in  time  between  New  York  and  St.  Louis  being  1  hr.  5 
min.  1  sec.  ? 

9.  The  difference  in  time  between  Philadelphia  and 
Chicago  is  49  min.  50  sec. ;  what  is  the  difference  in  lon- 
gitude ?     What  is  the  longitude  of  Philadelphia  ? 

10.  When  it  is  4  o'clock  (p.m.)  at  Greenwich,  what 
time  is  it  at  Washington  ? 

11.  When  it  is  1  o'clock  (a.m.)  at  New  York,  what  time 
is  it  at  Berlin  ? 


Art.  181.]  EXAMPLES.  171 

EXAMPLES  LXXII. 

Reduction  of  Metric  Numbers  to  Non-Metric  Numbers;    also,  of 
Non-Metric  Numbers  to  Metric  Numbers. 

1 .  How  many  cm  in  1  in.  ? 

2.  How  many  yd.  in  17.6 m  ? 

3.  How  many  t.  in  lMg  of  water  ? 

4.  How  many  sq.  ft.  in  lqm  ? 

5.  How  many  cu.  in.  in  l1  ? 

6.  How  many  lb.  in  lcum  of  water  ? 

7.  How  many  Mg  in  1  l.t.  ? 

8.  How  many  ml  in  1  qt.  (liquid)  ? 

9.  How  many  g  in  lb  5  ? 

10.  How  many  gr.  in  15 cg  ? 

11.  How  many  gr.  in  500 ccm  of  water  ? 

12.  How  many  HI  in  5  pk.  ? 

13.  How  many  bu.  in  3K1  ? 

14.  If  either  a  qt.  or  a  liter  of  milk  cost  6  ct.,  which 
would  you  prefer  to  purchase  ? 

15.  Which  would  you  prefer  to  buy,  1  A.  or  2.5  Ha  for 
the  same  money  ? 

16.  Find  the  value  of  3 13  in  g. 

17.  Find  the  value  of  lKg  in  lb. 

18.  Express  2  gal.   lpt.   3gi.   as  liters. 

19.  How  many  sters  in  100  cu.  ft.  ? 

20.  How  many  A.  in  7Ha  ? 

21.  Express  1-^-mi.  as  m  and  as  Hm. 

22.  What  cost  4  kilos  of  sugar  at  5±-  ct.  per  lb.  ? 

23.  What  costs  £  a  kilo  of  gold  at  $1  a  pwt.  ? 


172 


NON-DECIMAL   MEASURES. 


[Chap.  VI. 


Tables  for  Convenient  Reference. 


Time. 

Square  Measures. 

60  sec.    =  1  min. 

144  sq.  in.  =  1  sq.  ft. 

60  min.  =  1  hr. 

9  sq.  ft.  =  1  sq.  yd. 

24  hr.      =  1  da. 

30£  sq.  yd.  =  1  sq.  rd. 

365  da.     =  1  yr. 

160  sq.  rd.  =  1A. 

366  da.     =  1  leap  yr. 

640  A.        =lsq.mi. 

Troy  Weight. 

16  sq.  rd.  =  1  sq.  ch. 

10  sq.  ch.  =  1  A. 

24  gr.      =  1  pwt. 

20  pwt.   =  1  oz. 

Cubic  Measures. 

12  oz.      =  1  lb. 

1728  cu.  in.  =  1  cu.  ft. 

Avoirdupois  Weight. 

27  cu.  ft.  =  1  cu.  yd. 

16  dr.      =  1  oz. 

16  cu.  ft.  =  1  cd.  ft. 

16  oz.      =  1  lb. 

128  cu.  ft.  =  1  cd. 

100  lb.      as  1  cwt. 

20  cwt.   =  1 1. 

Liquid  Measures. 

112  lb.      =11.  cwt. 

4  gi.        =1  pt. 

22401b.      =ll.t. 

2  pt.         =1  qt. 

4  qt.        =1  gal. 

Apothecaries'  Weight. 

31£  gal.      =  1  bbl. 

gr.  20  =  3  1 

2  bbl.      =  1  hhd. 

3  s=  51. 

1  qt.        =  57|  cu.  in. 

3   8=  51. 

512  =  ft).  1. 

Dry  Measures. 

2  pt.        =1  qt. 

Linear  Measures. 

8  qt.        =1  pk. 

12  in.        =  1  ft. 

4  pk.       =  1  bu. 

3  ft.         =1  yd. 

1  qt.       =  67|  cu.  in. 

5^d'}    =  lrd. 
16^  ft.    i 

320  rd.        =  1  mi. 

Apothecaries'  Fluid 

Measures. 

ni60  =  f31. 
f38  =  f51. 

7.92  in.       =  1  li. 

100  li.         =  1  ch. 

f316  =  01. 

80  ch.       =  1  mi. 

08  =  Cong.  1. 

Art.  181.] 


SYNOPTIC  CONVERSION. 


173 


Synoptic  Conversion  of  English  and  Metric  Units.* 


English  to  Metric. 


lin. 
lyd. 
1  mi. 


a  2.54 cm. 
=  .9144m. 
=  1.60935  ki 


Metric  to  English. 

1™    =39.37  in. 

lKm  =  1093.61  yd. 
8Kra=5mi.  nearly. 


1  sq. 

yd. 

.83613  q«". 

Iqm 

=  lca  = 

10.7639  sq 

ft. 

1A. 

— 

.404687  Ha. 

1»    = 
lHa  = 

119.599  sq. 
2.471  A. 

yd. 

leu.  in.  =  16.3872"™ 
leu.  yd.  ss.  76456  c»m. 
lqt.  (U.  S.)=. 946361. 


leu  m_  61023.4  cu.  in. 
=  35.3145  cu.  ft. 
=  1.30794  cu.  yd. 

Icdm  1 

J,      |  =  61.023  cu.  in. 

==.26417  gal.  (U.S.) 
==  1.05668  qt.  (U.  S.). 


lgr.  =64. 7989  m*. 

1  lb.  avoir.  =  .45359*2. 

It.  (20001b.)     =907. 18 Kg. 
1  l.t.  (2240  lb.)  =  1.01605 t 


is  =  15.4324  gr. 

lKs  =  2.20462  lb.  avoir. 

IT  =  2204.62  lb.  avoir. 

IT  =.98421  l.t.  (2240  1b.). 


Weights. 

1  bu.  wheat 

=    60  lbs. 

1  stone                 =    14  lbs 

1   "    potatoes 

=    60    " 

1  bbl.  pork           =200    " 

1   "    beans 

=    60    " 

1   "     flour           =  196    " 

1   **    corn 

=    56    " 

1  cental  of  grain  =  100   " 

1    "    barley 

=    48    « 

1  quintal  of  fish   =100    " 

1   "    oats 

=    32    " 

*  Arranged  from  the  Smithsonian  Tables, 
black  type  should  be  memorized. 


Figures  printed  in 


174 


NON-DECIMAL   MEASURES.  [Chap.  VI. 


Inches. 


1     1 

,l,\l, 

il.'.l. 

.i.f.i, 

MM    Mil    IMI    Mill 

1 

!  } 

i 

i   i 

1         1 

3 

1 

i 

Centimeters. 
lin.  =  2.54cm. 


D 


l**1.  1  sq.  in. 

1  sq.  in.  =  6.45  icm. 


A 

/ 

> 

y 

lccm-  1  cu.  in. 

1  cu.  in.  =  16.387  ccm- 


Art.  181.] 


1T5 


Diameter  and  height  of  a  cylindrical  liter 
measure  and  of  a  cylindrical  quart  measure. 

li  =  61.063  cu.  in.  =  1000 ccm. 
1  qt.  =  57.75       " 


176  APPROXIMATION.  [Chap.  VII. 


CHAPTER  VII. 

APPROXIMATION. 

182.  No  continuous  magnitude  can  be  measured  with 
perfect  accuracy.  When,  for  example,  we  endeavor  to 
make  two  pieces  of  wire  equally  long,  all  that  we  can 
ensure  is,  that  they  shall  be  of  the  same  length  so  far  as 
the  eye,  or  other  instrument,  can  judge;  however,  they  may, 
and  probably  will,  differ  by  some  thousandths  or  even 
hundredths  of  an  inch. 

In  all  questions  involving  continuous  magnitude,  such 
as  length,  weight,  etc.,  we  must,  therefore,  be  content 
with  approximations  (more  or  less  accurate)  to  the  true 
measure.  It  follows  that  calculations  dependent  upon 
measurement  can  give  only  approximately  accurate 
results. 

For  example,  if  we  are  told  that  a  slab  of  stone  is  17.6  inches 
long,  and  12.4  inches  wide,  we  are  not  to  conclude  that  these  are 
perfectly  accurate  measurements,  but  only  that  the  measurements 
are  near  enough  for  practical  purposes,  the  real  length  and  breadth 
being  at  any  rate  less  than  17.7  and  12.5  respectively. 

If  the  given  measurements  were  accurate,  the  area  of  the  slab 
would  be  17.6  x  12.4  square  inches.  The  actual  area  may,  how- 
ever, have  any  value  between  17.6  x  12.4  square  inches  and 
17.7  x  12.5  square  inches  ;  that  is,  between  218.24  square  inches 
and  221.25  square  inches. 

183.  When  the  measure  of  any  quantity  is  given,  for 
example,  as  3.628,  it  generally  means  that  the  measure  is 


Arts.  182-184.]  APPROXIMATION.  177 

not  less  than  3.628,  and  not  greater  than  3.629,  the  possible 
error  made  by  stopping  at  the  third  decimal  place  being 
an  error  in  defect  less  than  one  one-thousandth  of  the 
unit.  Now,  if  the  above  measure  had  to  be  given  as  far 
only  as  hundredths  of  the  unit,  3.63  would  be  more  accu- 
rate than  3.62.  This  principle  is  often  employed  when 
approximate  measures  are  given.  Thus  the  quantity 
whose  measure  is  6.57684  would  be  most  accurately  given 
by  6.5768,  6.577,  or  6.58  to  four,  three,  or  two  decimal 
places  respectively,  the  possible  error  in  excess  or  defect 
being  now  not  greater  than  half  the  unit  represented  by 
the  last  decimal  place  retained. 

184.  To  find  the  sum  of  any  numbers  to  any  given  num- 
ber of  decimal  places,  it  would  be  necessary  to  consider 
the  figures  two  places  beyond,  in  order  to  see  what  had 
to  be  '  carried.' 

Ex.  1.   Find,    to   3  places  of  decimals,  the  sum  of   14.61825, 
3.17924,  .518479,  and  154.017235. 
14.618 


3.179 

.518 

154.017 

172.333 


25 
24 
479 
235 


Ex.  2.   Find,  to  within  one  one-thousandth  of  the  whole,  the  sum 
of  5.3184,  27.5162,  18.4196,  and  23.0135. 


5.31 
27.51 
18.41 
23.01 


74.27 


Here  we  have  to  find  the  sum  correct  to  the  first  four  figures. 
The  sum  of  the  numbers  in  the  fifth  column  is  27,  which  is  nearer 
to  30  than  to  20.  Hence,  the  most  accurate  sum  to  four  figures 
will  be  74.27. 


178 


APPROXIMATION. 


[Chap.  VII. 


185.  The  method  of  finding  a  product  or  a  quotient  to 
any  required  degree  of  accuracy  will  be  seen  from  the 
following  examples. 

Ex.  1.  Find,  to  two  places  of  decimals,  the  product  of  4.163  and 
5.784. 


4.10 

3 

5.7 

84 

20.81 

5 

2.91 

41 

.33 

30 

.01 

66 

24.08 

Arrange  with  the  decimal  point  of  the  multiplier  as  ahove,  and 
begin  the  multiplication  from  the  left  of  the  multiplier.  The  verti- 
cal line  on  the  left  gives  the  figures  which  are  to  be  finally  retained  ; 
it  is,  however,  necessary  to  go  two  places  beyond  to  see  what 
should  be  '  carried '  to  the  last  column  retained. 

Multiply  as  usual  so  long  as  all  the  figures  are  to  be  retained. 
In  the  present  case  all  the  figures  in  the  first  two  rows  are  to  be 
retained. 

Before  multiplying  by  8,  cross  out  the  last  figure  of  the  multipli- 
cand, namely  3  ;  then  multiply  416  by  8,  putting  down  the  first 
figure  of  the  product  (adding  in  mentally  what  would  be  carried 
from  the  multiplication  of  the  figure  crossed  out)  in  the  last 
column.  Now  cross  out  another  figure  of  the  multiplicand,  and 
multiply  what  remains  by  4,  again  putting  down  the  first  figure  of 
the  product  (with  what  must  be  carried  from  the  multiplication  of 
the  last  figure  crossed  out)  in  the  last  column.  Proceed  in  this 
way  to  the  end. 

Since  the  sum  of  the  figures  in  the  fifth  column  is  18,  the  most 
accurate  product  we  can  give  to  two  places  of  decimals  is  24.08. 


Ex.  2.  Find,  to  within  one  one-millionth 
of  the  whole,  the  product  of  51.6243  and 
112.4167. 


Here  we  have  to  find  the  product,  correct 
to  the  first  7  figures.      • 


61.622 
11 


5162.43 

516.243 

130.248 

20.649 

.516 

.309 

.036 


5803.433 


2.4 


8 

72 
24 
74 
13 


167 


Art.  185.] 


EXAMPLES. 


179 


Ex.  3.     Find,  to  within  one  one-millionth,  the  quotient 
516.24175  -*-  123.456. 
122.4^)516.24175(4.181585 
493  824 
22  4177 
12  3456 
10  07215 
9  87648 
19567 
12345 
7222 
6172 
1050 
987 
63 
61 

We  have  here  to  find  the  first  seven  figures  of  the  quotient.  Having 
found  the  first  three  figures  in  the  ordinary  way,  the  remaining  four 
figures,  being  less  by  two  than  the  number  of  figures  in  the  divisor, 
can  be  found  by  a  shortened  process ;  namely,  instead  of  annex- 
ing a  naught  at  every  stage  on  the  right  of  the  remainder  as  usual, 
we  strike  out  the  last  figure  on  the  right  of  the  divisor  instead, 
taking  care,  however,  to  use  the  last  figure  struck  out  to  see  what 
should  be  'carried'. 

Ex.  4.     Find,  to  the  nearest  penny,  the  value  of 

£51.3125  x  17.1874. 

Since  \d.  =  £  .001  nearly,  it  will  be  unnecessary  to  retain  more 

than  four  decimal  places  in  the  product. 

Thus,  &b\J8ffl 

17  .1874 


513.125 

359.1875 

5.1312 

4.1050 

.3591 

205 

5 
0 
8 
2 

'£881.9285 
20 

s.  18.5700 
12 

d.  6.84        Ans.  £881.  18s.  Id. 


180  APPROXIMATION.  [Chap.  VII. 

EXAMPLES   LXXIII. 
Written  Exercises. 

Find  the  following  to  the  nearest  thousandth  of  the 
whole : 

1.  14.625x31.857.  4.   138.714x89.47. 

2.  15.816  x  19.714.         •  5.   314.2108  -j-  18.306. 

3.  156.423x175.45.  6.   81.4623 -s- 129.54. 

7.  15.8193  x  6.7149  -j- 1.3425. 

8.  115.416  x  123.518  - 119.417. 

Find,  to  within  a  millionth  of  the  whole : 
9.    198.4653x5.194238.     10.    8.10976429-5-15.623. 

Find,  to  within  one  one-thousandth  of  the  whole,  the 
areas  of  the  rectangles  whose  dimensions  are : 

11.  17.215  in.  by  34.827  in. 

12.  184.27  yd.  by  112.53  yd. 

13.  Find,  to  4  places  of  decimals : 

(i)  1  +  i+    i    +^-L-n  + 


1  1x2  '  1x2x3  1x2x3x4 


1   1x2  1x2x31x2x3x4 

Find  the  value,  to  the  nearest  farthing,  of 

14.    £31.625x12.8743.    15.    £  119.48125  x  .46127. 

Find,  to  the  nearest  cent,  the  value  of 

16.  $15.23x18.24.  18.  $315.80x175.297. 

17.  $17.32x112.428.  19.   $30.47x2180.3079. 


Art.  185.]  MISCELLANEOUS   EXAMPLES.  181 

EXAMPLES   LXXIV. 

Miscellaneous  Examples,  Chapters  V,  VI,  VII. 

Written  Exercises. 

1.  Find  18  x  19  x  25  x  16f. 

2.  Express  .035,  .625,  .12288  as  common  fractions  in 
their  lowest  terms. 

3.  How  many  times  is  14  yd.  1  ft.  6  in.  contained  in 
244  yd.  3  in.  ? 

4.  Reduce  3  lb.  5  oz.  16  dwt.  to  gr.,  and  express  1  oz. 
16  dwt.  11  gr.  in  avoirdupois  weight. 

5.  Find  H.C.F.  and  L.C.M.  of  936  and  2925. 

6.  Arrange  -^,  yl^,  and  -fe  in  order  of  magnitude. 

7.  Find  the  cost  of  25cwt.   25  1b.   12  oz.  of  a  sub- 
stance at  $  16  per  cwt. 

8.  Find  the  value  of  51  things,  any  four  of  which  are 
worth  £  19.  3s.  Id. 

9.  Simplify  |f  (1  -  ff)  +  fx|(|  +  A)- 

10.  What  is  the  least  number  which  must  be  added  to 
1000000  that  the  sum  may  be  exactly  divisible  by  573  ? 

11.  Multiply  4  mi.  31  rd.  4|yd.  by  3,  and  divide  the 
result  by  37. 

12.  The  circumferences  of  the  large  and  the  small 
wheels  of  a  bicycle  are  143  in.  and  40  in.  respectively ; 
how  many  more  turns  will  the  latter  have  made  than 
the  former  in  a  distance  of  13  mi.  ? 

13.  A  man  spends  7.75  francs  a  day ;  how  much  does 
he  save  in  a  year  (of  365  days)  out  of  a  yearly  income 
of  3000  francs  ? 

14.  A  man  spends  9.35  marks  a  day;  how  much  in 
English  money  does  he  spend  in  a  year  (of  365  days), 
taking  a  mark  to  be  worth  ll|d  ? 


OF  THE     "^ 
I'NfVFDQlTV 


182  MISCELLANEOUS   EXAMPLES.       [Chap.  VII. 

15.  Afield  is  192 m  long  and  57.75 m  wide;  how  many 
Ha  does  it  contain,  and  what  wonld  it  cost  at  7500 
francs  per  Ha  ? 

16.  Eeduce  772642  sq.yd.  to  A.,  sq.  rd.,  and  sq.  yd. 

17.  Find,  in  hr.,  min.,  and  sec,  .6575  of  a  day. 

18.  What  fraction  of  8  lb.  11  oz.  2  dwt.  17  gr.  is  10 
lb.  9oz.  16  dwt.  11  gr.? 

19.  Reduce  ||,  -£fe,  and  flygft  to  decimals. 

20.  A  certain  number  was  divided  by  105,  by  'short' 
divisions ;  the  quotient  was  192,  the  first  remainder  was  1, 
the  second  was  4,  and  the  third  was  6.  What  was  the 
dividend  ? 

21.  Find  by  factors  the  square  root  of  23716. 


22.  What  is  the  greatest  sum  of  money  of  which  both 
$  11.05  and  $  188.50  are  multiples  ? 

23.  How  much  would  it  cost  to  put  gravel  to  a  depth 
of  a  dm  all  over  a  court-yard  7.5 m  by  5.75 m,  the  gravel 
and  labor  costing  8  francs  per  ster  ? 

24.  A  grocer  buys  15  cwt.  of  goods  for  $24.50;  at 
what  rate  per  lb.  must  he  sell  to  gain  $  5.50  ? 

25.  A  druggist  buys  50  1b.  of  a  certain  drug;  how 
many  weeks  will  it  last  if  he  uses  lb  1  5  6  3 1  3  2  gr.  10 
per  week  in  putting  up  prescriptions  ? 

26.  Find  If  of  8  bu.  1  pk. 

27.  How  many  numbers,  each  567,  must  be  added  that 
the  sum  may  be  greater  than  a  million  ? 

28.  What  is  the  greatest  number  of  Sundays  there  can 
be  in  a  year  ?  On  what  day  of  the  week  will  the  first  of 
February  fall  when  the  number  of  Sundays  in  a  year  of 
365  days  is  greatest  ? 


Art.  185.]  MISCELLANEOUS   EXAMPLES.  183 

29.  How  many  times  can  3  yd.  1  ft.  7  in.  be  sub- 
tracted in  succession  from  115  yd.  2  ft.  11  in.,  and  what 
will  be  the  last  remainder  ? 

30.  A  bar  of  metal  weighing  100  oz.  16  dwt.  is  made 
into  coins,  each  weighing  1  oz.  8  dwt. ;  how  many  coins 
are  made  from  the  bar  ? 


81.   Simplify  If  of  |L=^J2f-!  of -U. 

32.  A  surveyor  measured  some  ground  and  found  it  to 
be  10  ch.  long  and  4  ch.  broad ;  how  many  A.  were  there  ? 

33.  What  is  the  smallest  number  of  exact  acres  that 
can  have  the  form  of  a  square  ? 

34.  What  decimal  of  1  mi.  is  119yd.  2ft.  4 in.? 

35.  Find   the   value  of    21b.   6  oz.    10  dwt.    12  gr.    of 
gold  at  $  216  per  lb. 

36.  Find  105 2;  48  x  33£;  850  -  16|. 

37.  Express  lb.  1  as  the  decimal  of  1  lb.  Avoir. 

38.  Having  given  that  a  meter  is  39.37  in.,  prove  that 
the  difference  between  5  mi.  and  8Km  is  nearly  51  yd. 


39.  Add 

i  °f  feJ  °f  • 4-65  to  ft-  °f  3*3  °f  • Li5- 

°3  %  +  *  ±J-TT  ^  —  lS 

40.  Find  Vlwt an(^  re(iuce  the  answer  to  lowest  terms. 

41.  Express  .88125  cwt.  in  lb.  and  oz. 

42.  Express  15  yd.  2  ft.  8  in.  as  the  decimal  of  a  mi. 

43.  Reduce  11.2765625  lb.  to  lb.,  oz..  pwt.,  and  gr. 

44.  Find  to  the  nearest  cent  $  48.96  x  72.8967. 

45.  Reduce  1000  sq.  yd.  to  qm. 

46.  Reduce  1000 l  to  pt. 


184         MISCELLANEOUS  EXAMPLES.     [Chaps.  VII.,  VIII. 

47.  Express  .136  x  7.3  -s-  .43  as  a  decimal. 

48.  Find  the  value  of  43  sq.  rd.  24^  sq.  yd.  of  building 
land  at  $  1815  per  acre. 

49.  Find  the  greatest  length  of  which  both  1  mi.  4  fur. 
16  rd.  2  yd.  and  1  mi.  1  fur.  10  rd.  2  yd.    are   multiples. 

50.  Subtract  161  x      **     from    5f  of  7\  . 

51.  Find  y.004  to  4  decimal  places. 

52.  Reduce  4Hg  to  pounds  Troy. 

7 


53.    Simplify 


5  + 


3-f 


54.  Find  the  annual  cost  of  repairing  a  road  9  mi. 
120  rd.  177  yd.  long  at  $  88  per  mi. 

55.  A  vessel  steams  18  knots  an  hour;  to  how  many 
statute  miles  is  this  equivalent  ? 

56.  If  a  ccm  of  iron  weighs  7.788g,  what  will  be  the 
weight  of  a  cu.  ft.  ? 

57.  How  many  pieces  each  .17  in.  long  can  be  cut  from 
a  wire  21.09  in.  long ;  and  how  long  will  be  the  piece 
left  over? 

58.  Add  .5125  of  a  yd.,  .62734  of  a  rd.,  and  .018325  of 
a  fur. ;  subtract  the  result  from  .0049  of  a  mi.,  and  ex- 
press the  answer  in  yd.,  also  in  dm. 

59.  Find  V4900546043.21156004. 

60.  What  is  the  least  number  which  when 
divided  by  15  leaves  a  remainder  3,  when 


it 

«  18 

u 

u 

a 

6, 

(( 

"  24 

a 

a 

a 

12? 

Arts.  186,  187.] 


AREAS. 


185 


CHAPTER  VIII. 

AREAS  —  VOLUMES. 

186.  A  plane  figure  [Art.  149]  bounded  by  four  straight 
lines,  and  whose  four  angles  are  equal,  is  called  a 
Rectangle. 

An  equilateral  rectangle  is  a  Square. 


Rectangles. 


Square. 


The  amount  of  surface  included  within  the  bounding 
lines  of  a  figure  is  called  its  Area,  and  the  area  is  measured 
by  some  square  unit,  —  one  sq.  in.,  one  sq.  yd.,  or  one 
qm,  etc. 


B 
C 

A 

G 
D 

F 

B 


187.  To  find  the  Area  of  a  Rectangle.  —  Let  ABCD  be 

the  rectangle  whose  area  is  required. 


186  AEE AS  — VOLUMES.  [Chap.  VIII. 

Suppose,  for  example,  that  AB  is  4  in.  and  that  AD  is  3  in. 
Divide  AB  into  four  equal  parts  and  AD  into  three  equal  parts, 
and  draw  lines  parallel  to  the  sides  as  in  the  figure  on  the  left. 

Then  the  rectangle  is  divided  into  squares  each  of  which  is  a 
sq.  in. ;  and  the  number  of  these  squares  is  clearly  the  product 
of  the  number  of  in.  in  AB  by  the  number  of  in.  in  AD. 

The  above  reasoning  applies  to  all  cases,  both  the  length  and 
the  breadth  of  the  rectangle  being  an  integral  number  of  in. 

Now  suppose,  for  example,  that  in  the  figure  on  the  right  AB  is 
§  in.,  and  that  AD  is  £  in. 

Let  AEFG  be  one  sq.  in.  Divide  AE  into  two  equal  parts, 
and  AG  into  five  equal  parts,  and  GD  into  two  equal  parts. 
Then  the  subdivisions  of  AB  will  be  all  equal,  as  also  those  of  AD. 
Hence,  if  lines  be  drawn  as  in  the  figure,  ABCD  will  be  divided 
into  3x7  equal  rectangles,  such  that  the  square  inch  AEFG  will 

3x7 
contain  2  x  5  of  these  rectangles.     Hence  AB  will  contain 

square  inches  ;  that  is,  (§  x  |)  square  inches. 

From  the  above  it  follows  that  the  number  of  square 
inches  (or  square  feet,  etc.)  in  a  rectangle  is  equal  to  the 
product  of  the  number  of  inches  (or  feet,  etc.)  in  the  length 
by  the  number  of  inches  (or  feet,  etc.)  in  the  breadth. 

It  should  be  noticed  that  the  length  and  breadth  must 
both  be  expressed  in  terms  of  the  same  unit. 

For  example,  the  area  of  a  rectangle  whose  length  is  2  ft.  and 
breadth  6  in.  is  (2  x  T62)  sq.ft ,  or  (24  x  6)  sq.  in. 

The  above  rule  for  finding  the  area  of  a  rectangle  is  often  ex- 
pressed shortly  by  the  statement  that  area  =  length  x  breadth. 

188.  Now  that  we  find  the  area  of  a  rectangle,  we  can 
see  that  the  relations  between  the  different  units  given  in 
the  Table  for  Square  Measure,  on  page  149,  follow  at  once 
from  the  relations  between  the  corresponding  units  in 
linear  measure. 

For,  since  12 in.  make  1  ft.,  (12  x  12)  sq.in.  make  1  sq.ft. 
Since  3  ft.  make  1yd.,  (3  x  3)  sq.ft.  make  1  sq.  yd. 


Arts.  188,  189.]  EXAMPLES.  187 

Since  5|yd.  make  1  rd.,  (5£  x  5|)  sq.  yd.  make  1  sq.  rd. 

Again,  22  yd.  make  1  ch.,  therefore  (22  x  22)  sq.yd.  =  484  sq. 
yd.  make  1  sq.  ch. 

Thus,  4840  sq.  yd.  =  10  sq.  ch.  =  1  A. 

Also,  1  sq.  mi.  =  (1760  x  1760)  sq.  yd.  =  1760  x  1760  «■  4840  A. 
=  640  A. 

Ex.  Find  the  acreage  of  a  rectangular  field  whose  length  is  132 
yd.  and  whose  breadth  is  38 1  yd. 

The  area  =  (132  x  38£)  sq.  yd. 

=  5082  sq.yd.  =  ffff  A. 
=  |iA.  =  1A.  8sq.  rd. 

189.  If  the  area  of  a  rectangle  be  known,  and  also  the 
length,  the  breadth  can  be  at  once  found. 

For  example,  to  find  the  breadth  of  a  rectangle  whose  length  is 
15  ft.  and  whose  area  is  200  sq.  ft. 

Since  the  product  of  the  number  of  ft.  in  the  breadth  by  the 
number  of  ft.  in  the  length  is  equal  to  the  number  of  sq.  ft.  in 
the  area,  we  have 

breadth  =  (200  +  15)  ft.  =  13£  ft.  =  13  ft.  4  in. 


EXAMPLES  LXXV. 
Written  Exercises. 

Find  the   areas   of  the  rectangles  whose  lengths  and 
breadths  are  as  follows : 

1.  14  ft.,  12  ft.  6.  10  yd.,  23  ft. 

2.  22  ft,  17it.  7.  5  yd.  lft.,  3  yd.  2  ft. 

3.  25  yd.,  17  yd.  8.  21  yd.  2  ft.,  18  yd. 

4.  122  in.,  114  in.  9.  13  ft.  4  in.,  9  ft.  2  in. 

5.  5  ft.,  17  in.  10.  11  ft.  9  in.,  8  ft.  7  in. 


188  AREAS— VOLUMES.  [Chap.  VIII. 

Find  the  acreage  of  the  rectangular  fields  whose  lengths 
and  breadths  are  as  follows : 

11.  319  yd.,  275  yd.  15.  550  yd.,  400  yd. 

12.  363  yd.,  240  yd.  16.  125  yd.,  49£  yd. 

13.  400  yd.,  214|-yd.  17.  Length  x  breadth  =  ? 

14.  178|yd.,  162fyd.  18.  Area -i- length  =  ? 

19.   Area  -r-  breadth  =  ? 

20.  Find  the  area  of  a  rectangular  field  whose  length 
is  119.5 m  and  whose  breadth  is  96.2 m. 

21.  How  many  stones  having  rectangular  tops  2dm 
X  1.2 dm  will  be  required  to  pave  a  street  5Hm  long  and 
16.8 m  wide,  provided  no  spaces  are  left  between  the 
stones  ? 

22.  Find  the  area  of  a  rectangular  field  whose  length 
is  9  ch.  12  li.  and  whose  breadth  is  6  ch.  25  li. 

23.  Find  the  area  of  a  rectangular  field  whose  length 
is  9  ch.  25  li.  and  whose  breadth  is  7  ch.  75  li. 

24.  The  area  of  a  rectangle  is  925  sq.  in.,  and  its 
breadth  is  25  in. ;  what  is  its  length  ? 

25.  What  is  the  length  of  a  rectangular  table  the  area 
of  whose  top  is  71  sq.  ft.  16  sq.  in.,  and  the  breadth  6  ft. 
8  in.? 

26.  The  area  of  a  rectangular  court-yard  is  52  sq.  yd. 
2  sq.  ft.  36  sq.  in.,  and  its  length  is  14  yd.  9  in. ;  what  is 
its  breadth  ? 

27.  What  will  it  cost  to  paint  the  ceiling  of  a  room 
whose  length  is  24  ft.  6  in.  and  breadth  16  ft.  6  in.  at 
$.60  per  sq.  yd.  ? 

28.  What  is  the  area  of  a  square  floor  7m  long  ? 

29.  What  is  the  length  of  a  square  room  whose  area  is 
4225  qdm? 


Art.  190.]  CARPETING,  ETC.  189 


Carpeting,  Papering,  Plastering. 

190.  Examples  like  the  following  are  of  frequent 
occurrence : 

Ex.1.  How  much  will  be  the  cost  of  a  cm-pet  for  a  room  16  ft. 
x  20  ft.  3  in.  with  carpet  27  in.  wide  at  $.75  a  yd.,  the  strips  running 
lengthwise  ? 

Number  of  strips  =  20  ft.  3  in.  +-  27  in.  =  9. 

Total  length  of  carpet  =  16  ft.  x  9  =  144  ft.  =  48  yd. 
Cost  =  48  yd.  x  $.75  =  $36.     [Art.  50.] 

Ex.  2.  How  much  will  be  the  cost  of  paper  for  the  walls  of  a 
room  19/£.  3  in.  long,  15/£.  9  in.  wide,  and  12  ft.  high,  the  paper 
being  21  in.  wide  and  costing  5  ct.  per  yard  ? 

Area  of  a  wall    =  its  length  x  its  height. 
.  •.  Area  of  4  walls  =  distance  around  the  room  x  height 
=  70  ft.  x  12  ft. 
=  840  sq.  ft. 
Length  of  paper  =  840  sq.  ft.  -r-  f  \  ft. 

=  480  ft  =  160  yd. 
Cost  of  paper      =  160  yd.  x  5  ct. 
=  $8.00. 

Note.  In  the  preceding  questions  we  have  found  the  quantity 
of  carpet  (or  wall  paper)  which  would  be  required  if  it  were  of 
one  uniform  color  throughout.  When,  as  is  almost  invariably  the 
case,  there  is  a  pattern  on  the  carpet  or  paper,  there  must  be  a 
certain  amount  of  waste,  if  the  different  lengths  are  properly 
fitted  together.  Moreover,  wall  papers  are  sold  in  lengths  of  8 
yards,  called  rolls ;  if,  therefore,  as  in  Ex.  2,  160  yards  of  paper 
were  required,  20  rolls  would  have  to  be  bought.  American  wall 
papers  are  generally  18  inches  wide. 

Ex.  3.  A  room  21  ft.  by  19/£.  has  a  Turkey  carpet  in  it,  a  border 
3ft.  wide  all  round  being  left  uncovered  by  the  carpet.  The 
border  was  stained  at  a  cost  of  $  .45  a  square  yard,  and  the  carpet 
cost  $4.50  a  square  yard;  what  was  the  total  cost  ? 

Since  the  border  is  3  ft.  wide  all  round  the  room,  the  length  of 
the  carpet  must  be  21  ft.  -  3  ft.  x  2  =  15  ft.,  and  the  breadth  must 
be  19  ft.  -  3  ft.  x  2  =  13  ft. 


190  AREAS  — VOLUMES.  [Chap.  Vlll. 

Area  of  carpet    =  15  ft.  x  13  ft.  =  -6/  sq.  yd. 

Price  of  carpet  =  $4.50  x  6^-  =  $97.50. 

Border  =  area  of  room  —  area  of  carpet 

a=  (21  x  19)  sq.  ft.  -  (15  x  13)  sq.ft. 

as  204  sq.  ft.  ss  -638-  sq.  yd. 
Cost  of  staining  =  $.45  x  -^  =  $10.20. 
Total  cost  =  $  97. 50  +  $  10.20 

=  $107.70. 


EXAMPLES  LXXVI. 
Written  Exercises. 

1.  How  much  carpet  27  in.  wide  will  cover  a  room 
22  ft.  6  in.  long  and  15  ft.  9  in.  wide,  carpet  running 
lengthwise  ?     What  will  be  the  cost  at  $1.20  per  yd.  ? 

2.  A  room  is  8.3 m  long  and  5m  wide;  how  many 
meters  of  carpet  must  be  purchased  for  such  a  room,  the 
strips  being  7dm  wide  and  running  crosswise  ?  How  much 
in  width  must  be  turned,  under  ?     In  surface  ? 

3.  If  you  were  carpeting  a  room  9m  x6m,  which  way 
would  you  have  the  strips  run  if  they  were  6.8 dm  wide? 
How  many  less  qm  would  be  used  than  by  running  the 
strips  the  other  way  ? 

4.  A  room  is  10  yd.  2  ft.  long  and  7  yd.  1ft.  6  in. 
wide ;  find  the  cost  of  covering  it  with  Turkey  carpet  at 
$1.25  a  sq.  yd. 

5.  Find  the  cost  of  carpeting  a  room  8^  yd.  long  by 
6  yd.  2  ft.  broad  with  carpet  2\  ft.  wide  at  84  ct.  a  yd. 

6.  What  would  be  the  expense  of  carpeting  a  room 
24  ft.  6  in.  by  18  ft.  with  carpet  27  in.  wide,  and  which 
costs  $1.20  a  yd.  ? 


Art.  190.]  CARPETING,  ETC.  191 

7.  How  much  carpet  27  in.  wide  would  be  required 
for  a  room  32  ft.  by  23  ft.,  a  margin  4  ft.  wide  being 
left  uncovered  ? 

8.  Find  the  area  of  the  four  walls  of  a  room  15  ft. 
long,  14  ft.  wide,  and  10  ft.  high. 

9.  Find  the  area  of  the  four  walls  of  a  room  16  ft. 
4  in.  long,  13  ft.  8  in.  wide,  and  11  ft.  4  in.  high. 

10.  Find  the  area  of  the  four  walls  of  a  room  10.5  ■ 
long,  5ra  wide,  and  4.9 m  high. 

11.  Find  the  qm  of  the  four  walls  of  a  room  7m  x  4m 
X  3.2 ra,  leaving  out  3  windows,  each  2m  x  1.1 m,  and  one 
door  2.4m  x  1.3m. 

12.  Find  the  area  of  the  four  walls  of  a  room  14  ft. 
6  in.  long,  13  ft.  10  in.  wide,  and  10  ft.  8  in.  high. 

13.  A  room  is  18  ft.  long,  13  ft.  6  in.  wide,  and  12  ft. 
high ;  how  much  paper  21  in.  wide  will  be  required  to 
cover  the  walls,  and  what  will  be  the  cost  at  $  .75  per 
piece  of  12  yd.  ? 

14.  How  much  will  it  cost  to  paper  a  room  17  ft.  6  in. 
square,  and  14  ft.  3  in.  high,  with  paper  1  ft.  9  in.  wide 
at  12  ct.  a  yd.  ? 

15.  A  room  is  6.1ra  x  5m  x  4.2m;  find  the  cost  of  plas- 
tering at  62^- ct.  per  qm,  allowing  7Qm  for  windows, 
door,  and  base-board.     Do  not  forget  the  ceiling. 

16.  A  room  8m  x  5.6m  x  4.2 m  has  4  windows,  each 
2.1 m  x  lm,  2  doors,  each  2.8 m  x  1.4 m,  and  a  base-board 
24 dm  high;  find  (i)  the  cost  of  plastering  at  50  ct.  per 
qm,  (ii)  the  cost  of  paper  5dm  wide  at  $3.50  per  roll  of 
10 m,  (iii)  the  cost  of  a  carpet  6.2 dm  wide  at  $2  per  m, 
all  for  this  room.  Find  the  total  cost,  allowing  $  15  for 
labor  in  putting  on  the  paper  and  laying  the  carpet. 


192  AREAS  — VOLUMES.  [Chap.  VIII. 


Board  Measure. 

191.  A  board  which  is  one  foot  square  and  one  inch  or 
less  in  thickness  has  a  measurement  called  one  Board 
Foot. 

Boards  and  squared  timber  are  sold  by  the  Board  Foot. 

192.  The  number  of  board  feet  in  a  board  one  inch  or 
less  in  thickness  is  the  same  as  the  number  of  square  feet 
in  the  surface. 

The  number  of  board  feet  in  a  stick  of  timber  more 
than  one  inch  thick  is  the  number  of  square  feet  in  the 
surface  multiplied  by  the  number  of  inches  in  the  thick- 
ness. 

Ex.  1.  How  many  board  feet  in  a  board  20/£.  x2ft.  x  f  of 
an  inch  f 

20  ft.  x  2  f t.  =  40  board  feet. 

Ex.  2.  How  many  board  feet  in  a  board  15  ft.  long,  IS  in.  wide 
at  one  end  and  14  in.  wide  at  the  other  end,  and  \  an  inch  thick  ? 

15  ft.  x  1 J  ft.  =  20  board  ft. 

In  this  case  the  average  width  is  used. 

Ex.  3.  How  many  board  feet  in  a  stick  of  timber  21.6  ft.  long, 
14  in.  wide,  and  3§  in.  thick  ? 

21.6ft.  x  lift.  =  25.2  board  ft., 

if  the  stick  were  1  in.  or  less  in  thickness.     But  we  must  multiply 
this  result  by  3|,  since  the  timber  is  3|  in.  thick.    Thus, 

21.6  ft.  x  |  ft.  x-:f  =  92.4  board  ft. 

EXAMPLES  LXXVII. 
Written  Exercises. 

Find  the  number  of  board  feet  in  the  following : 

1.  A  board  20  ft.  x  2  ft.  x  1£  in. 

2.  A  board  19  ft.  8  in.  x  1  ft.  9  in.  x  £  in. 


Arts.  191-193.]    DIMENSIONS   OF   CIRCLES.  19S 

3.  A  timber  13  ft.  x  1.1  ft.  x  4J  in. 

4.  A  joist  11  ft.  x  5  in.  x  2  in. 

5.  A  joist  16  ft.  x  6  in.  x  2\  in. 

6.  Find  the  cost  of  each  of  the  above  five  pieces  at 
$20  per  M;  i.e.f  by  the  thousand  (board  feet). 

7.  A  hall  75  ft.  x  50  ft.  has  two  layers  of  boards  for 
its  floor,  one  kind  costing  $  12  per  M,  and  the  other  cost- 
ing $21  per  M;  the  floor  timbers,  70  in  number,  are 
placed  crosswise,  and  cost  $  16  per  M.  How  much  is  the 
cost  of  material,  the  boards  being  fin.  thick,  and  the 
timbers  8  in.  x  3  in. 

Dimensions  of  Circles. 

193.  Cut  from  cardboard  a  circular  piece  having  a 
known  radius,  as  3cm,  or  3  in.  Roll  the  circle  (held 
upright)  along  a  straight  line  and  measure  the  line  trav- 
ersed in  one  complete  rotation  of  the  circle.  This  line 
will  be  found  to  be  about  3^  times  the  diameter. 

We  have  no  means  of  finding  the  exact  measure  of  the 
circumference  in  terms  of  the  diameter,  but  by  means  of 
geometry  we  learn  that  the  measure  is  3,1416  (nearly) 
times  the  diameter.     This  is  more  exact  than  3^. 

Hence,     Diameter  x  3.1416  =  Circumference, 
and  C  -f-  3.1416  =  D.* 

Using  the  results  obtained  in  geometry  for  areas  of 
circles,  we  have, 

Area  =  R2  x  3.1416, 

R2     =  Area      --  3.1416, 
R       =  VArea  -h  3.1416. 

*  Z),  B,  and  C  stand  for  diameter,  radius,  and  circumference, 
respectively. 


194  AREAS  —  VOLUMES.  [Chap.  VIII. 

Ex.  1.    The  diameter  of  a  circle  is  10dOT;  find  the  circumference 
and  area. 

C=Dx  3.1416  Area  =  i?2  x  3.1416 

=  10 dm  x  3.1416  =  25^  x  3.1416 

=  31.416dm.  =  78.54  «*". 

Ex.  2.    The  area  of  a  circle  is  50.2656  sq,  in. ;  find  the  radius 
and  the  circumference. 


JR=  VArea-s-3.1416  C  =  D  x  3.1416 


=  V50.2656  *■  3.1416  =  8  x  3.1416 

s  Vl6  =25.1328  in. 

=  4  in. 

EXAMPLES  LXXVIII. 
Written  Exercises. 

Find  the  circumference  when 

1.  D  =  14  in.  3.    E  =  15cm.  5.    Z)  =  56yd. 

2.  2)  =  75  ft.  4.    B  =  l$m.  6.    iJ  =  -J-mi. 
Find 

7.    R  when  0  =  314.16 cm.       8.    D  when  C  =  1  mi. 
9.    D  when  O  =  153.9384  yd. 
10.    iZwhen  C  =  47.124 m. 

Find  area  when 

11.  5  =  14ft  13.    C=37.6992dm. 

12.  Z)  =  20m.  14.    C  =  251.328  rd. 

15.  How  many"  sq.  ft.  in  the  floor  of  a  circular  room 
whose  diameter  is  28  ft.  ? 

16.  The  bottom  of  a  round  liter  measure  has  a  surface 
of  500 qcm;  find  the  approximate  radius. 

17.  Find  the  cost  of  concreting  a  circular  fountain 
basin  whose  diameter  is  20  ft.,  the  work  and  material 
costing  f  3.27  per  sq.  yd. 


Arts.  194,  195.]       RECTANGULAR  SOLIDS. 


195 


Rectangular  Solids. 

194.  That  which  has  length,  breadth,  and  thickness  is 
called  a  Solid. 

A  solid  bounded  by  six  rectangular  [Art.  186]  faces  is 
called  a  Rectangular  Solid. 

A  cube  [Art.  151]  is  one  form  of  a  rectangular  solid. 

Any  substance  (water,  air,  wood,  etc.)  may  be  a  rectan- 
gular solid  in  form. 

The  space  included  between  the  bounding  surfaces  of 
a  solid  is  called  its  Capacity  (or  Volume),  and  the  capacity 
of  a  solid  is  measured  by  some  cubic  unit — :Oiie  cu.  in., 
one  cu.  ft.,  lccm,  or  lcum,  etc. 


195.  To  find  the  Capacity  of  a  Rectangular  Solid.' 
Suppose,  for  example,  that  the  dimensions  of  the  solid 
are  5  in.  by  4  in.  by  3  in.  We  can  divide  the  edges  re- 
spectively into  5,  4,  and  3  parts,  each  being  one  inch ;  and 
if  planes  be  drawn  through  the  points  of  division  parallel 
to  the  outer  faces  of  the  solid,  as  in  the  figure,  the  whole 
solid  will  be  divided  into  equal  cubes  each  of  which  is  a 
cubic  inch. 


There  will  be  as  many  layers  of  cubes  as  there  are  inches 
in  the  height  of  the  solid,  and  the  number  of  cubes  in  each 
layer  will  be  the  product  of  the  number  of  inches  in  the 
length  by  the  number  of  inches  in  the  breadth. 


196  AREAS  — VOLUMES.  [Chap.  VIII. 

Thus,  the  number  of  cubic  inches  (or  cubic  feet,  etc.)  in  a 
rectangular  solid  is  equal  to  the  continued  product  of  the 
number  of  inches  (or  feet,  etc.)  in  its  length,  breadth,  and 
thickness. 

196.  Now  that  we  can  find  the  capacity  of  a  rectangular 
solid,  we  can  find  the  relations  between  the  cubic  yard, 
the  cubic  foot,  and  the  cubic  inch. 

For  1  cu.  yd.  =  (3  x  3  x  3)  cu.  ft., 

and  1  cu.  ft.    =  (12  x  12  x  12)  cu.  in. 

Ex.  1.  Find  the  volume  of  a  rectangular  block  of  stone  12  ft. 
long,  1ft.  wide,  and  lft.  6  in.  high. 

Volume  =  (12  x  7  x  If)  cu.  ft.  =  126  cu.  ft. 

Ex.  2.  A  beam  lft.  6 in.  wide  and  lft.  Sin.  high  contains  46\ 
cubic  feet  of  timber  ;  what  is  its  length  ? 

Since  volume  =  length  x  breadth  x  thickness, 

lengths volume -• 

breadth  x  thickness 

4g  i 

Hence,  length  required  = 1 —  241  ft. 

\\  x  l£ft.         3 

Ex.  3.  How  many  gallons  of  water  will  a  cistern  hold  if  it  is 
6  ft.  long,  4  ft.  6  in.  wide,  and  3  ft.  6  in.  high  ?  [A  gallon  con- 
tains 231  cu.  in.~] 

The  cistern  will  hold 

(72  x  54  x  42)  cu.  in.  =  163296  cu.  in. 

Hence,  the  number  of  gallons  required  =  163296  -4-  231  =  706.90. 

Ex.  4.  The  external  dimensions  of  a  rectangular  stone  tank  are  : 
length  12  ft.  6  in.,  breadth  8  ft.,  and  height  4  ft.  The  interior  is  also 
rectangular,  and  the  sides  and  bottom  are  3  in.  thick.  Find  the 
number  of  cu.ft.  of  stone  in  the  tank. 

The  internal  length    =  12  ft.  6  in.  -  3  in.  x  2  =  12  ft., 
the  internal  breadth        =    8  ft.  -  3  in.  x  2  =    7  ft.  6  in., 

and  the  internal  height  «=    4  f t.  —  3  in.         =    3  ft.  9  in. 


Art.  196.]  EXAMPLES.  197 

Now  the  volume  of  the  stone  is  the  difference  between  the 
volumes  given  by  the  external  and  internal  dimensions. 
Hence,  volume  required 

=  (12$  x  8  x  4  -  12  x  7£  x  3|)  cu.  ft. 
=  (400  -  337i)  cu.  ft.  =  62£  cu.ft. 

EXAMPLES   LXXIX. 
Written  Exercises. 

Find  the  volumes  of  the  rectangular  solids  whose 
dimensions  are 

1.  5  ft.  by  4  ft.  by  2  ft. 

2.  12  ft.  by  6  ft.  by  4  ft. 

3.  3  yd.  by  l^yd.  by  2  ft. 

4.  5  yd.  by  21  yd.  by  4  ft. 

5.  6  ft.  4  in.  by  4  ft.  3  in.  by  2  ft.  6  in. 

6.  7  ft.  9  in.  by  5  ft.  3  in.  by  3  ft.  6  in. 

7.  5  yd.  1ft.  by  3  yd.  2  ft.  by  2  ft.  9  in. 

8.  6  yd.  9  in.  by  2  yd.  1ft.  by  2  ft.  7  in. 

9.  A  rectangular  block  of  stone  4  ft.  long  and  2  ft. 
6  in.  broad  contains  17-J  cu.  ft.  of  stone ;  what  is  its 
height? 

10.  Find  the  length  of  a  rectangular  beam  which  con- 
tains 98  cu.  ft.  of  timber  and  whose  cross-section  is  2  ft. 
square. 

11.  How  many  loads  (cu.  yd.)  of  gravel  would  be 
required  to  cover  a  path  150  yd.  long  and  4  ft.  wide  to 
a  depth  of  2  in.  ? 

12.  A  school-room  whose  floor  is  60  ft.  by  40  ft.  has 
accommodation  for  360  children,  allowing  100  cu.  ft.  of 
air  for  each  child;  what  must  be  the  height  of  the  room  ? 


198  AREAS  — VOLUMES.  [Chap.  VIII. 

13.  If  1  gal.  =  231  cu.  in.  and  1  gal.  of  water  weighs 
8.355  lb.  Avoir.,  find  the  number  of  gal.  and  the  weight 
of  the  water  which  would  fall  on  an  area  of  an  A.  during 
a  rainfall  of  one  in. 

14.  A  tank  is  21ft.  4  in.  long,  3  ft.  wide,  and  2  ft. 
deep ;  it  is  filled  with  water  to  within  3  in.  of  the  top. 
What  is  the  volume  of  the  water,  and  what  is  its  weight  ? 
[A  cu.  ft.  of  water  weighs  1000  oz.] 

15.  What  weight  of  water  will  fall  on  a  road  \  a  mi. 
long  and  30  ft.  wide  during  a  rainfall  of  an  in.  ? 

16.  A  level  tract  of  land  20  mi.  long  and  f  of  a  mi. 
broad  is  flooded  to  a  depth  of  4  ft.  Given  that  a  cu.  ft. 
of  water  weighs  62.5  lb.,  find  in  t.  the  weight  of  the 
water  on  the  land. 

17.  What  is  the  capacity  of  a  tank  20m  x  8m  x  2m? 
How  many  T  of  water  will  it  hold  ?  Eeduce  the  T  to 
t.  (tonneaux  to  tons). 

18.  Find  the  total  surface  of  the  stone  in  Ex.  9. 

19.  Find  the  inner  surface  of  the  tank  in  Ex.  17. 

20.  A  square  room  is  5m  long  and  3m  high;  how  many 
cu.  in.  of  air  will  the  room  contain  ? 

21.  A  square  room  9  ft.  3  in.  high  has  a  capacity  of 
1563 \  cu.  ft.  j  what  is  the  length  of  the  room  ? 

Cylinders. 

197.  A  solid  whose  ends  are  circles  and  whose  curved 
surface  is  perpendicular  to  the  ends  is  called  a  Right 
Circular  Cylinder. 

The  ends  are  called  Bases, 

For  example,  a  common  lead  pencil  is  a  right  circular  cylinder  ; 
and  some  tin  measures  used  for  liquids  are  right  circular  cylinders. 

Note.  When  cylinders  are  mentioned  in  this  book,  right  circu- 
lar cylinders  are  meant. 


Arts.  197,  198.]  CYLINDEKS.  199 

198.   The  total  surface  of  a  cylinder  consists  of  two 

flat  surfaces  (circles) 
and  a  curved  surface 
called  the  Lateral  Sur- 
face. 

If  a  piece  of  paper  be 
fitted  to  a  cylinder  so  as 
to  cover  all  its  lateral  sur- 
face and  then  unrolled,  it 
will  be  a  rectangle  whose 
length  is  the  circumference 
of  the  cylinder  and  whose 
breadth  is  the  height  of  the 
cylinder. 

Hence,  lateral  surface  =  Cx  height ; 
which  (by  Art.  193)        =  D  x  3.1416  x  H; 
whence  H=  lateral  surface  -r-  (D  x  3.1416), 

and  D  =  lateral  surface  -r-  (3.1416  x  H). 

Ex.  1.    Find  the  total  surface  of  a  cylinder  8  in.  high  and  the 
radius  of  whose  base  is  2  in. 

Total  surface  =  2  bases  +  lateral  surface 

=  2  iF2  x  3.1416  +  D  x  3.1416  x  H 
=  2B  x  3.1416  (22 +  #) 
=  4  x  3.1416  x  10 
=  125.664  sq.  in. 

Note.    H  =  height ;  D  =  diameter  ;  C  =  circumference. 

Ex.  2.    The  lateral  surface  of  a  cylinder  is  188.496 idm  ;  find  the 
height  when  D  =  6. 

H=  138.496  -  (D  X  3.1416) 
=  188.496  -4-  18.8496 
=  10  ^ 


200 


AREAS— VOLUMES. 


[Chap.  VIII. 


199.  If  a  cylinder  is  10  in.  high,  it  is  evident  that  it 
will  contain  10  times  as  many  cubic  inches  las  if  it  were  1 
in.  high;  since  the  number  of  cubic 
inches  in  a  cylinder  1  in.  high  is  the 
same  as  the  number  of  square  inches  in 
the  base,  the 

Volume  of  a  cylinder  =  base  x  H, 
or  (Art.  193),  =  R2  x  3.1416  x  H, 
Volume 


and 


and 


and 


H= 


R2  = 


R2  x  3.1416' 

V 
5.1416  x  H' 


*=v 


3.1416  x  H 


Ex.  1.    Find  the  volume  of  a  cylinder  whose  height  is  10  in.  and 
the  radius  of  whose  bast  is  6  in. 

V=B*x  3.1416  xH 
=  36  x  3.1416  x  10 
=  1130. 976  cu.  in. 

Ex.  2.    Find  the  radius  of  the  base  of  a  cylinder  whose  volume 
is  125.664 cdm  and  whose  height  is  lm. 


JB 


-V 

=  ^V  3.1416  x  10 
b  2dm. 


3.1416  x  H 
125.664 


EXAMPLES  LXXX. 
Written  Exercises. 


Find,  in  a  cylinder,  the 

1.   Lateral  surface  when  R  =  3dm  and  H=  14 dm. 

3.   Lateral  surface  when  R  =  1  in.  and  H  =  5  in. 


Arts.  199,  200.]  SPECIFIC   GRAVITY.  201 

3.  Total  surface  when  D  =  10m  and  H=  8m. 

4 .  H  when  lateral  surface  =  1413.72  idm  and  D  =  15 dm. 

5.  Fwheni2  =  3dm   and   i7=14dm. 

6.  V  when  R  ==  1  in.  and  H=  5  in. 

7.  FwhenD  =  10m  and   JfiT=8m. 

8.  IT  when  V=  125.664  cu.  ft.  and  B  =  2  ft. 

9.  jRwhen  F=1570.81  and  17  =  20dm. 

10.  Measure  in  centimeters  the  height  and  diameter  of 
some  cylinder  and  calculate  how  many  cubic  centimeters 
of  liquid  it  would  hold  if  hollow.  How  many  grams  of 
water  would  it  hold  ? 

Specific  Gravity. 

200.  Weigh  accurately  a  stone.  Then  place  it  in  a  jar 
brimful  of  water  and  weigh  the  water  which  runs  over. 
Now  divide  the  weight  of  the  stone  by  the  weight  of  the 
water  which  ran  over,  and  you  will  know  how  many  times 
the  weight  of  the  stone  is  greater  than  the  weight  of 
the  same  volume  of  water. 

The  number  of  times  that  the  weight  of  a  substance  is 
greater  than  the  weight  of  the  same  volume  of  water  is 
called  the  Specific  Gravity  (S.G.)  of  the  substance. 

A  body  floating  in  water  displaces  a  weight  of  water 
equal  to  its  own  weight. 

EXAMPLES  LXXXI. 
Written  Exercises. 

1.  The  S.G.  of  iron  is  7.8 ;  how  much  does  a  cu.  ft.  of 
iron  weigh  ?     A  cu.  in.  ?     A  ccm  ? 

2.  What  is  the  weight  of  a  cdm  of  silver,  its  S.G.  being 
10.5? 


202  AREAS  — VOLUMES.  [Chap.  VIII. 

3.  A  rectangular  iron  tank  weighs  25  kilos,  and  it 
floats  on  water;  what  is  the  weight  of  the  water  dis- 
placed ?  What  is  the  volume  of  water  displaced  ?  What 
is  the  volume  of  iron  in  the  tank  ? 

4.  A  cubical  liter  measure  weighs  150  grams ;  if  put 
in  water,  what  pressure  must  be  added  to  its  own  weight 
to  make  it  sink  ? 

5.  The  S.G.  of  gold  is  19.5;  if  a  person  can  lift  125 
lb.,  how  many  cu.  in.  of  gold  can  he  lift  at  one  time  ? 

6.  What  is  the  weight  of  a  cdm  of  gold  in  pounds 
Avoir.  ?  In  pounds  Troy  ?  What  is  its  value  at  $1  per 
pwt.  ? 

If  the  stone  mentioned  in  Art.  200  be  weighed  in  air 
and  then  in  water,  the  loss  of  weight  will  be  found  equal 
to  the  weight  of  the  water  which  ran  over.  Therefore 
if  we  divide  the  weight  of  a  substance  by  its  loss  of  weight 
in  water,  we  shall  obtain  its  S.G. 

7.  A  substance  weighs  2501b.  in  air  and  1251b.  in 
water ;  what  is  its  S.G.  ?  How  much  water  would  run  over 
if  the  substance  were  put  into  a  jar  brimful  of  water? 
What  is  the  volume  of  the  substance  ? 

8.  A  piece  of  wood,  S.G.  .25,  floats  on  water  and  dis- 
places 40  ccm  of  water ;  what  is  the  volume  of  the  wood  ? 
How  much  iron  must  be  attached  to  the  wood  to  make  it 
float  under  water  ? 

9.  How  much  weight  will  a  ccm  of  iron  lose  when 
weighed  in  air  and  then  in  water  ? 

Note.   S.G.  =  weight  in  air  -f-  loss  in  water. 

10.  A  person  weighing  146^  lb.  has  a  S.G.  of  1.0417 ; 
how  much  does  he  weigh  in  water  ? 


Art.  200.]         MISCELLANEOUS   EXAMPLES.  203 

EXAMPLES   LXXXII. 
Miscellaneous  Examples,  Chapter  VIII. 

1.  Find  the  cost  of  graveling  a  carriage  drive  69  ft. 
9  in.  long  and  16  ft.  wide,  at  30  ct.  per  sq.  yd. 

2.  The  outer  and  inner  boundaries  of  a  gravel  path 
are  squares,  and  the  path  is  4  ft.  wide.  The  side  of  the 
square  enclosed  by  the  path  is  50  yd.  How  much  would 
it  cost  to  gravel  the  path  at  37^  ct.  a  superficial  yd.  ? 

3.  Find  the  cost  of  turfing  a  lawn  tennis  court  which 
is  78  ft.  long  and  39  ft.  wide,  making  a  margin  of  grass 
12  ft.  wide  at  each  end  and  6  ft.  wide  at  each  side ;  the 
turf  costing  4d.  a  sq.  yd. 

4.  Find  the  prime  numbers  from  100  to  125  by  using 
the  sieve.     [Art.  78.] 

5.  What  will  be  the  lowest  cost  of  carpeting  a  room 
33  ft.  long  and  24  ft.  wide  with  carpet  27  in.  wide  and 
costing  85  ct.  a  yd.,  a  border  one  yd.  wide  being  left 
uncovered  ?  How  broad  a  strip  must  be  cut  off,  or  turned 
under  ? 

6.  Multiply  688.4  by  99;  460.01237  by  11. 

7.  Find  the  number  of  cu.  ft.  in  a  school-room  by  using 
its  length,  breadth,  and  height ;  find  also  the  area  of  its 
six  surfaces,  including  windows,  etc. 


8.  Find  the  cost  of  covering  the  floor  of  a  hall  39  ft. 
41  in.  long  by  20  ft.  llf  in.  wide  with  tiles  each  5J  in. 
by  4|in.,  and  costing  (including  the  labor  of  laying 
them)  $4.95  a  hundred. 

9.  What  is  tile  weight  of  an  iron  girder  20  ft.  long, 
and  having  54  sq.  in.  sectional  area,  the  weight  of  iron 
being  480  lbs.  per  cu.  ft.  ? 


204  MISCELLANEOUS   EXAMPLES.     [Chap.  VIII. 

10.  A  hall  is  103.23  ft.  long  and  83.25  ft.  broad,  and 
it  is  to  be  paved  with  equal  square  tiles ;  what  is  the  size 
of  the  largest  tile  which  will  exactly  fit,  and  how  many 
of  them  will  be  required  ? 

11 .  A  room  22  ft.  3  in.  long,  17  ft.  9  in.  wide,  and  12  ft. 
6  in.  high  has  two  windows  each  5  ft.  3  in.  by  3  ft.  4  in., 
a  door  7  ft.  by  3  ft.  9  in.,  and  a  fireplace  5  ft.  3  in.  by 
4  ft.  4  in.  How  many  pieces  (each  12  yd.  long)  of 
paper  21  in.  wide  would  have  to  be  bought  to  paper  the 
room? 

12.  How  many  sq.  ft.  of  boards  are  required  for  the 
floor  of  a  circular  hall  100  ft.  in  diameter  ? 

13.  What  would  be  the  weight  of  a  beam  of  oak  5dem 
square  and  7m  long,  on  the  supposition  that  the  S.G-.  of 
oak  is  .895  ? 

14.  Divide  8921045  by  385  by  the  method  of  Art.  69. 


15.  Supposing  a  postage  stamp  to  be  1  in.  long  and 
|  of  an  in.  broad,  how  many  stamps  would  be  required 
to  cover  a  wall  which  is  15  ft.  6  in.  long  and  10  ft.  8  in. 
high? 

16.  What  is  the  cost  of  paper  for  the  hall  of  Ex.  12, 
the  hall  being  25  ft.  high,  at  $.57  a  roll,  allowing  500 
sq.  ft.  for  windows,  etc.  ? 

17 .  A  cubical  cistern,  open  at  the  top,  costs  £16. 13s.  4d. 
to  line  with  lead  at  2d.  per  qdm;  how  many  cum  of 
water  will  it  hold  ? 

18.  I  have  a  rectangular  box  cover  \  of  a  meter  long 
and  A  of  a  meter  broad  to  be  painted  in  squares ;  what 
is  the  largest  square  I  can  use  ? 

19.  Find    v  to  three  decimal  places. 


Art.  200.J         MISCELLANEOUS   EXAMPLES.  205 

20.  Find  the  number  of  board  feet  in  a  stick  of  timber 
8 \  in.  square  at  one  end,  8%  in.  by  5}  in.  at  the  other, 
and  21  ft.  long. 

21.  Find,  by  dividing  by  factors,  40579  -r-  72. 

22.  How  many  cdm  of  water  are  required  to  fill  a 
cylindrical  tank  whose  radius  is  10dm  and  whose  height 
is2ra? 

23.  A  cylindrical  tank  holds  1884.96 \  and  its  height 
is  16f  dcm;  what  is  its  radius? 

24.  How  many  cu.  in.  of  wood  are  there  in  a  wooden 
box  whose   external   dimensions  are   4  ft.  4  in.   by   3  ft. 

10  in.  by  3  ft.  6  in.,  the  wood  being  everywhere  1  in. 
thick  ? 

25.  An  iron  safe  is  everywhere  l^in.  thick,  and  its 
external  dimensions  are  6  ft.  by  4  ft.  6  in.  by  3  ft.  6  in. 
How  much  does  the  iron  weigh?  [The  S.G.  of  iron  has 
been  given  several  times.] 

26.  It  cost  81.75/.  to  gravel  a  rectangular  court-yard 
8.25 m  x  4.8  m  with  gravel  costing  7.5/.  per  st.  What  was 
the  thickness  of  the  layer  of  gravel  ? 

27.  An  importer  received  427  T  of  goods  at  125/  per 
T;  he  paid  the  custom  house  $1602.92;  his  expenses 
of  cartage,  etc.,  were  $212 ;  for  how  much  must  he  sell 
the  goods  per  cwt.  in  order  to  make  a  profit  of  $2000  ? 

28.  The  velocity  of  flow  of  water  through  a  pipe  6cm  in 
diameter  is  7.6 **  per  sec;  how  many  1  flow  through 
in  11  sec.  ? 

29.  A  room  is  18  ft.  1  in.  long,  lift.  8  in.  wide,  and 

11  ft.  3  in.  high ;  how  many  rolls  of  paper  would  be  used 
in  papering  the  walls,  supposing  that  windows,  etc.,  make 
up  i  of  the  whole  surface  of  the  walls  ? 


206  MISCELLANEOUS   EXAMPLES.      [Chap.  VIII. 

30.  How  much  would  it  cost  to  carpet  the  room  of  Ex. 
29,  the  carpet  being  1  yd.  wide,  at  $1.66§  per  yd.  ? 

31.  The  S.G-.  of  a  piece  of  wood  is  .5;  the  wood  being 
8m  x  8dm  x  8cm,  how  many  kilos  would  be  required  to  sink 
the  wood  if  placed  on  water  ? 

32.  Find  J\  2  |600  -  (4  150  -  25.5  -  16)  +  3} 

-  (4.1  12-2)  -  (*i  x  90  +  50 


33.  The  S.G.  of  gold  is  19.5;  find  the  weight  of  lcdm 
and  the  volume  of  1  kilo. 

34.  A  pile  of  wood  is  8  ft.  long,  4  ft.  high,  and  4  ft. 
broad ;  what  is  its  volume  in  cu.  ft.  ? 

35.  How  many  such  volumes  (Ex.  34)  in  a  pile  49.6 
ft.  long,  8ft.  high,  and  4ft.  thick? 

36.  A  quadrangle  120  ft.  x  100  ft.  has,  in  the  center,  a 
grass-plot  80  ft.  x  60  ft. ;  find  the  cost  of  graveling  the 
rest  of  it  to  a  depth  of  6  in.  at  54  ct.  a  cu.  yd. 

37.  At  a  certain  place  the  annual  rainfall  was  24.15 
in. ;  find  the  number  of  gal.  which  fell  on  each  sq.  mi. 

38.  A  slate  cistern  open  at  the  top  is  everywhere  1 
in.  thick,  and  the  external  dimensions  are  length  6  ft. 
4  in.,  breadth  3  ft.  2  in.,  and  height  4  ft.  8  in.  Find  the 
weight  of  the  slate  employed,  assuming  that  1  cu.  ft.  of 
slate  weighs  2880  oz. 

39.  Find  the  number  of  gal.  the  cistern  in  the  previous 
question  will  hold. 

40.  What  is  the  height  of  a  cylindrical  liter  measure 
if  the  radius  of  its  base  is  5cm  ? 

Answer  to  the  nearest  tenth  of  a  mm. 


Art.  200.]  MISCELLANEOUS  EXAMPLES.  207 

41.  State  the  squares  of  23,  34,  38,  47,  78,  and  96, 
using  the  method  indicated  in  Arts.  54  and  86. 

42.  Find  by  factors  the  H.C.F.  and  L.C.M.  of  168, 
2772,  4368,  and  12474. 

43.  Find  the  weight  of  12 m  of  alcohol,  its  S.G.  being 
.81. 

44.  A  room  is  24  ft.  2  in.  x  18  ft.  11  in. ;  which  way 
would  it  be  the  cheaper  to  run  the  carpet  strips,  each 
strip  being  27  in.  wide  ? 

45.  Find  the  dividend  when  the  divisor  is  yf  and  the 
quotient  is  J4. 


208  RATIO  —  PROPORTION.  [Chap.  IX. 


CHAPTER  IX. 

RATIO  —  PROPORTION. 

201.  The  quotient  of  one  number  divided  by  another 
of  the  same  kind  may  be  called  the  Ratio  of  the  first  to 
the  second. 

Thus,  the  ratio  of  6  ft.  to  2  ft.  is  3,  and  the  ratio  of  62  cwt.  to 
16  cwt.  is  ||,  or  3|. 

A  ratio  is  expressed  by  the  sign  :  placed  between  the 
two  quantities ;  this  sign  means  the  same  as  the  sign  -j-. 

Thus,  52  m  :  26  m  means  52  m  -=-  26  » 

The  first  is  read  '  the  ratio  of  52  m  to  26 m '  ;  the  second  is  read 
'the  quotient  obtained  by  dividing  52 m  by  26 m.'  The  answer  is 
the  same  in  both  cases.     A  ratio  may  be  also  expressed  in  the  form 

52  m 

of  a  fraction  ;  as, 

'  26m 

202.  The  two  quantities  compared  in  a  ratio  are  called 
the  terms  of  the  ratio. 

The  old  names,  antecedent  and  consequent,  for  the  first  and 
second  terms  respectively  of  a  ratio,  are  still  sometimes  used. 

The  terms  of  a  ratio  must  be  of  the  same  kind  of 
magnitude;  for  we  cannot  compare,  for  example,  tons 
with  weeks,  or  acres  with  gallons. 

When  a  ratio  and  one  of  its  terms  are  given,  the  other 
term  can  at  once  be  found. 


Akts.  201, 202.]  EXAMPLES.  209 

Ex.  1.    A  ratio  is  3,  and  its  first  term  is  6  ;  find  its  second  term. 

Here  a  dividend  6  and  a  quotient  3  are  given ;  hence,  the  divisor 
=  6-7-3  =  2  =  the  second  term  of  the  ratio. 

Ex.  2.    A  ratio  is  4.5,  and  its  second  term  is  10 ;  find  its  first 
term. 

Here  a  quotient  4.5  and  a  divisor  10  are  given  ;    hence,  the 
dividend  =  4.5  x  10  =  45  =  the  first  term  of  the  ratio. 

Ex.  3.     What  sum  of  money  has  to  $30  the  ratio  of  5:8? 

Here  the  ratio  is  §,  and  its  second  term  is  $30 ;  hence,  $30  x  $ 
=  $18.75  =  the  first  term. 


EXAMPLES    LXXXIII. 
Oral  Exercises. 

Find  the  indicated  ratios,  and  in  lowest  terms  : 

1.  9:3;  16:4;  50:5;  20:9;  12:16. 

2.  18:30;  75:100;  90:30;  .5:5;  5 :  .5. 

3.  121b.  :51b.;   20  gr.  :  33;    $ 5  :  $ 55 ;  $ .50  :  $5.50. 

4.  24g:36g;  50cum:20cum. 

5.  A  square  2  ft.  long :  a  square  1  ft.  long. 

6.  A  cube  2  in.  long  :  a  cube  1  in.  long. 

7.  A  circle  6m  in  diameter  :  a  circle  2m  in  diameter. 

8.  What  is  the  ratio  of  a  square  to  another  square 
half  as  long?  Twice  as  long?  One-third  as  long? 
Three  times  as  long? 

9.  What  is  the  ratio  of  a  cube  to  another  cube  half 
as  long  ?    Twice  as  long  ? 

10.   What  is  the  ratio  of  a  circle  to  another  circle 
having  twice  the  diameter  ?    Three  times  the  diameter  ? 
Five  times  ? 
p 


210  RATIO  — PROPORTION.  [Chap.  Dt. 

Written  Exercises. 

Find,  in  lowest  terms,  the  indicated  ratios : 

11.  $5  :  $7.50 ;  $3.25  :  $12.50 ;  $5.44  :  $39.10. 

12.  3t.5  cwt.  :  It.  15  cwt.;  3cwt.  641b.  :  4cwt.  761b. 

13.  7  mi.  208  rd.  :  4  mi.  277  rd. ;  2  oz.  11  pwt.  18  gr. : 
51    32    &1   gr.l. 

14.  Find  what  has  to  $1.20  the  ratio  of  2  :  3. 

15.  Find  what  has  to  1  da.  4hr.  20  min.  the  ratio  of 
3:4. 

16.  Find  what  has  to  11  cwt.  55  lb.  the  ratio  of  7  :  11. 

17.  Find  what  has  to  £  11.  14s.  9d.  the  ratio  of  2. 

18.  Find  the  second  term,  the  first  term  being  4  cd. 
24  cu.  ft.,  and  the  ratio  being  f,  or  8  :  9. 

Proportion. 

203.   When  fonr  quantities  are  such  that  the  ratio  of 

the  first  to  the  second  is  equal  to  the  ratio  of  the  third 

to  the  fourth,  the  four  quantities  are  said  to  be  Propor- 
tionals. 

For  example,  the  ratio  $5:  $15  =  the  ratio  3t.  :  9t.  Hence, 
the  four  quantities,  $5,  $15,  3 1.,  and  9t. ,  are  proportionals. 
The  notation  is  as  follows  : 

$5  :  $15  =  3 1.  :  9 1. ; 
or,  $5  :  $15  : :  3 1.  :  9  t. 

This  is  read,  '$5  is  to  $15  as  3t.  is  to  9t.'  ;  meaning  that  the 
ratio  between  $5  and  $15  is  the  same  as  that  between  3 1.  and  9 1. 

A  proportion  may  be  expressed  in  the  form 

$5  =  3  t. 
$15     9t.' 


Arts.  203-205.]  PROPORTION.  211 

It  follows  that,  when  two  fractions  are  equal,  the  terms 
of  the  fractions,  taken  in  the  order  in  which  they  are 
written,  are  proportionals. 

Since  the  two  terms  of  any  ratio  must  be  of  the  same 
kind  of  magnitude,  the  first  and  second  terms  of  a  propor- 
tion must  be  of  the  same  kind,  and  the  third  and  fourth 
terms  must  be  of  the  same  kind. 

204 .*  The  first  and  fourth,  of  four  quantities  in  pro- 
portion, are  called  the  Extremes,  and  the  second  and 
third  are  called  the  Means. 

In  the  above  case,  the  ratios  are  T\  and  |  respectively ;  and 
since  T57  =  §,  it  follows  that  5  x  9  =  15  x  3. 

Thus,  in  this,  and  similarly  in  other  cases,  the  product  of  the 
extremes  is  equal  to  the  product  of  the  means. 

205.  When  any  three  terms  of  a  proportion  are  known, 
the  remaining  term  can  be  found. 

Ex.  1.     What  quantity  :  18  lb.  :  :  $ 4  :  $12  ? 
For  convenience,  let  x  stand  for  the  quantity  to  be  found  ;  then, 
x  lb.  :  18  lb. :  :  $4  :  $12, 
whence  [Art.  204]      12  x  x  =  18  x  4. 

.  •.  once  x  =  18  x  T%  =  6  lb. 
This  is  equivalent  to  saying  that  T4^  is  a  ratio,  and  18  lb.  is  its 
second  term  ;     the  first  term    (or  dividend)    must  be    18  x  TV 
[Art.  58;  also  202,  Ex.2.] 

Ex.  2.    27  oz.  :  15  oz.  :  :  what  quantity  :  25  in.  f 
27  oz.  :  15  oz  ;  :  x  in.  :  25  in. 
15  x  x  =  27  x  25 ; 
once  x  =  f|  x  25 

=  45  in.  =  Ans. 

*  It  will  be  noticed  that,  in  any  proportion,  when  the  first  term 
is  larger  than  the  second,  the  third  is  larger  than  the  fourth  ;  that, 
when  the  first  is  smaller  than  the  second,  the  third  is  smaller  than 
the  fourth. 


212  RATIO  —  PROPORTION.  [Chap.  IX. 

We  observe  from  the  above  that 

either  extreme  =  P^ct  of  means . 
the  other  extreme 

Likewise,     either  mean      =  P^ct  of  extremes, 

the  other  mean 

206.  When  the  second  term  equals  the  third  term,  we 
have  but  three  different  quantities  in  the  proportion ;  the 
second  is  then  called  a  Mean  Proportional  between  the  first 
and  third,  and  the  third  is  called  a  Third  Proportional 
to  the  first  and  second. 

Thus,  in  4  :  6  : :  6  :  9,  6  is  a  mean  proportional  between  4  and  9, 
and  9  is  a  third  proportional  to  4  and  6. 

Here  9  _  square  of  the  second  and  fl  =  ./product  of  the  extremes. 

the  first 


EXAMPLES   LXXXIV. 

1 .  Find  the  fourth  proportional  to  4,  7,  and  12. 

2.  Find  the  fourth  proportional  to  9,  8,  and  3. 

3.  Find  the  fourth  proportional  to  -1-,  ^,  and  f. 

4.  Find  the  mean  proportional  between  4  and  9. 

5.  Find  the  mean  proportional  between  8  and  18. 

6.  Find  the  mean  proportional  between  |  and  f. 

7.  What  quantity  has  to  $1.32  the  same  ratio  that 
4  ft.  3  in.  has  to  2  ft.  9  in.  ? 

8.  To  what  sum  has  Is.  9d  the  same  ratio  as  6  days 
10  hr.  has  to  7  days  8  hr.  ? 

9.  Fill  up  the  blank  in  the  proportion 

£  1.  12s.  6d.  :  £  2.  2s.  6d  :  :  1  cwt.  18  lb.  : . 

Questions  in  which  a  missing  term  of  a  proportion  has  to 
be  found,  and  other  questions  of  a  similar  nature,  are  best 
treated  by  a  method  which  we  now  proceed  to  consider. 


Arts.  206,  207.]     THE   UNITARY  METHOD.  213 


The  Unitary  Method. 

207.   The   method   will   be   seen   from   the  following 
examples : 

Ex.  1.    If  5  lbs.  of  tea  cost  $2.75,  how  much  will  8  lbs.  cost  at  the 
same  rate  ? 

Since  cost  of  5  lbs.  =  $2.75, 

«     "  1  lb.    =  $2.75  -  5. 
.-.  "     "  8  lbs.  =  $2.75  -5x8  =  $4.40. 

Ex.  2.   If  lcwt.  24lb.  of  sugar  cost  $8.06,  how  much  will  2cwt. 
46  lb.  cost  at  the  same  rate? 

Since        cost  of  lcwt.  241b.  (1241b.)  =$8.06, 
the  "     "  lib.  =$8.06-124 

=  $.065, 
and  "     "2  cwt.  46  lb.  (246  lb.)  =  $.065  x  246 

=  $15.99. 

Ex.  3.   How  long  would  24  horses  take  to  consume  the  same 
quantity  of  food  that  45  horses  eat  in  16  days  ? 

Since     45  horses  eat  the  food  in  16  days, 

45  x  16  horses  would  eat  the  food  in  1  day  ; 
.-.  45  x  16-24      "  "  "        "        24  days. 

Thus,  the  number  of  days  required  =  45  x  16  —  24  =  30. 


EXAMPLES   LXXXV. 
Oral  Exercises. 

1.  If  5t.  cost  $35,  what  will  20 1.  cost  at  the  same 
rate? 

2.  A  man  walked  12  mi.  in  3hr. ;  how  far  would  he 
walk  in  1\  hr.  ? 

3.  A  certain  quantity  of  food  would  be  consumed  by 
18  persons  in  15  da. ;  how  long  would  it  last  90  persons  ? 


214  RATIO  — PROPORTION.  [Chap.  IX. 

4.  If  18  yd.  are  bought  for  $3.60,  how  much  will  45 
yd.  cost  ? 

5.  How  far  should  15 1.  be  carried  for  the   money 
charged  for  carrying  12  t.  5  mi.  ? 

Written  Exercises. 

6.  If  27  men  can   mow  a  field   in   8  hr.,   how  long 
will  36  men  take  to  mow  the  same  field  ? 

7.  If  18  yd.  are  bought   for   $16.50,  find  the  price 
of  111  yd. 

8.  How  far  should  100  t,  be  carried   for  the  money 
charged  for  carrying  75  t.  a  distance  of  120  mi.  ? 

9.  If  19  men  do  a  certain  piece  of  work  in  117  da., 
how  long  will  it  take  13  men  to  do  the  same  work  ? 

10.    If  19  horses  can  be  bought  for  $475,  how  many 
can  be  bought  for  $700  at  the  same  rate  ? 

11. .  If  25  cows  cost  $1387.50,  how  much  will  6  cost  at 
the  same  rate  ? 

12.  If  a  train  runs  704  yd.  in  12  sec,  how  long  will 
it  take  to  go  half  a  mi.  ? 

13.  How   many   men  would   do   in   20  da.  the   same 
amount  of  work  as  15  men  can  do  in  16  da.  ? 

14.  How  long  would  75  horses  take  to  consume  the 
same  quantity  of  food  that  40  horses  eat  in  15  da.  ? 

15.  If  I  lend  a  man  $100  for  14  weeks,  how  long  ought 
he  to  lend  me  $175  in  return  ? 

16.  If  7cwt.  41b.  of   steel   cost  $133.76,  what  will 
3  cwt.  4  lb.  cost  at  the  same  rate  ? 

17.  If  1  cwt.  19  lb.  of  coffee  cost  $41.65,  how  much 
will  5  cwt.  cost  at  the  same  rate  ? 


Art.  208.]  EXAMPLES.  215 

18.  If  123  yd.  of  silk  cost  $165.05,  how  much  can 
be  bought  for  $58.05  at  the  same  rate? 

19.  A  man  walks  9  mi.  in  2  hr. ;  how  long  will  he  take 
to  walk  12  mi.  at  the  same  rate  ? 

20.  If  I  lend  a  man  $350  for  34  weeks,  how  long  ought 
he  to  lend  me  $170  in  return  ? 

21.  If    gold    is    worth   $18.60   per  oz.,   what    is    the 
value  of  a  cup  weighing  7  oz.  5  dwt.  12  gr.  ? 

22.  Find  the  value  of  12  things  any  7  of  which  are 
worth  $26.46. 

23.  If  3^- lb.  can  be  bought  for  $5.46,  how  much  can 
be  bought  for  $26.52  ? 

24.  If  a  t.  of  sugar  cost  $110,  how  much  will  8  cwt. 
26  lb.  cost  at  the  same  rate  ? 

208.  Each  of  the  examples  in  the  last  exercise  may  be 
solved  by  the  method  of  Art.  205. 

For  instance,  in  7,  the  price  of  18  yd.  holds  the  same  ratio  to  the 
price  of  111  yd.  that  18  holds  to  111 ;  hence, 
18  :  111 :  :  $16.50  :  x  ; 
x  =  $16.50  x  -W- 
=  $101.75. 
Here  TXT8T  is  the  ratio  and  $16.50  is  the  first  term  ;  therefore,  we 
must  divide  $16.50  by  ^  to  find  the  second  term. 

Again,  in  6,  the  time  required  for  36  men  is  f|  (ratio)  of  the 
time  required  for  27  men  ;  therefore, 

27  :  36  :  :  x  :  8  ; 
x  =  ft  X  8 
=  6  hours. 

EXAMPLES  LXXXVI. 
Written  Exercises. 

Perform  examples  8-15  in  the  last  exercise,  using  the 
method  of  Art.  208. 


216  RATIO  — PROPORTION.  [Chap.  IX. 


Similar  Figures  —  Similar  Solids. 

209.  Figures  or  Solids  which  have  the  same  shape  are 
called  Similar  Figures  or  Similar  Solids. 

In  the  cases  of  rectangular  figures,  and  rectangular 
solids,  and  cylinders  [note,  Art.  197],  sameness  in  shape 
is  determined  by  the  ratios  which  exist  between  lines 
having  the  same  relative  positions.  If  these  ratios  are 
equal,  the  figures  or  solids  have  the  same  shape. 

For  instance,  two  rectangles,  12  ft.  and  8  ft.  in  length,  and  3  ft. 
and  2  ft.  in  height,  are  similar,  because  12  ft. :  8  ft. :  :  3  f t.  :  2  ft. 

Likewise,  two  cylinders,  15 dm  and  9dm  in  height,  and  10 cm  and 
6cm  in  diameter,  are  similar,  because  15 dm :  9dm  : :  10 cm  :  6m. 

From  preceding  examples  we  have  learned  that  heights 
of  squares  are  proportional  to  their  lengths;  that  circum- 
ferences of  circles  are  proportional  to  their  diameters;  that 
surfaces  of  squares  or  circles  are  proportional  to  the  squares 
of  lengths  or  diameters;  and  that  volumes  of  cubes  are  pro- 
portional to  cubes  of  lengths  or  heights  or  breadths. 

What  is  true  of  squares,  circles,  and  cubes,  is  true  of 
all  similar  figures ;  viz., 

(1)  Lines  are  proportional  to  lines  (height  to  length,  etc.) ; 

(2)  Surfaces  are  proportional  to  squares  of  corresponding 
lines ; 

(3)  Volumes  are  proportional  to  cubes  of  corresponding 
lines. 

EXAMPLES  LXXXVII. 
Written  Exercises. 

1 .  The  circumference  of  a  circle  is  12  in. ;  what  is  the 
circumference  of  a  circle  whose  diameter  is  3  times  as 
great  ? 


Arts.  209,  210.]       PROPORTIONAL   PARTS.  217 

2.  A  rectangle  16  m  x  10 m  is  similar  to  another  rec- 
tangle whose  length  is  4m;  what  is  the  height  of  the 
second  rectangle  ? 

3.  What  is  the  area  of  a  rectangle  5  in.  long  when  a 
similar  rectangle  9  in.  long  has  an  area  of  32.4  sq.  in.  ? 

4.  What  are  the  comparative  areas  of  two  similar 
figures  whose  lengths  are  8cm  and  17 cm? 

5.  Two  cubes  are  13 m  and  lm  long;  how  large  is  the 
first  in  terms  of  the  second  ? 

6.  Two  similar  cylinders  have  diameters  of  5dm  and  3dm 
respectively :  compare  their  lateral  surfaces ;  their  bases  ; 
their  volumes. 

7.  A  cylindrical  bin  will  hold  300  bu.  of  wheat;  a 
similar  one  3  times  as  high  will  hold  how  many  bu.? 

8.  A  cylinder  2m  high  and  9dm  in  diameter  will  hold 
how  many  kilos  of  water  ?  What  will  be  the  diameter 
of  a  similar  cylinder  which  will  hold  10178784 g? 

Proportional  Parts. 
210.   Partnership. 

When  the  ratio  between  the  parts  of  a  given  quantity 
are  known,  the  parts  themselves  can  be  at  once  found. 

Ex.  1.  Divide  $100  between  A  and  B  so  that  A  may  have  $3 
for  every  $2  that  B  has. 

For  every  $3  that  A  receives,  B  will  receive  $ 2,  and  the  two 
together  will  receive  $3  +  $2  =  $5. 

Hence,        A  receives  $3  out  of  every  $5  of  the  whole  ; 
.  •.  A       "        |  of  the  whole  =  §  of  $100  =  $60. 

Also,  B        "        |  of  the  whole  =  f  of  $100  =  $40. 

Ex.  2.  The  profits  of  a  business  are  to  be  divided  between  the 
partners  A,  B,  and  C,  so  that  A  may  have  4  parts,  B  3  parts,  and 
C  2  parts.     How  much  does  each  get  out  of  a  profit  of  $4500  ? 


218  RATIO  —  PROPORTION.  [Chap.  IX. 

If  A  has  4  parts  to  B's  3  parts  and  C's  2  parts,  A  ivill  have  4 
parts  out  of  (4  +  3  +  2)  parts  divided  between  them  all. 

Hence  A  will  have of  the  whole  ; 

4  +  3  +  2 

.-.  A  will  have  $  of  $4500  =  $2000. 

B     «      «     f  of  $  4500  =  $  1500, 

and  C     "      «|  of  $4500  =  $  1000. 

Ex.  3.   Divide  $23.50  between  A,  B,  and  C,  so  that  A's  share 
may  be  to  B's  share  as  4  :  5,  and  B's  share  to  C's  share  as  3  :  4. 

Here  A's  share  =  f  of  B's  share,  and  B's  share  =  f  of  C's  share  ; 
.-.A's      "     =  f  of  f  of  C's  share  =  f  of  C's  share. 

Hence  A,  B,  and  C  have  together  (f  +    f  +  1)  of  C's  share  ; 


i.e., 

$23.50  = 

.12  +  15  +  20 

20 

C's 
$10. 

share  = 

=  t? 

of 

$23.50  = 

:$10 

;    A'S  =  |   Of    I 

Or  thus : 

A 

's  share 

:B's 

=  4:5, 

B's: 

C's=       3:4. 

Now  multiply  the  terms  of  the  two  ratios  by  such  numbers  that 
the  numbers  corresponding  to  B's  share  may  be  the  same  in  both. 

In  the  present  case,  multiply  by  3  and  5  respectively.    Then 

A's  share  :  B's  :  C's  =  12  :  15  :  20. 
Thus,  A  gets  12  parts  out  of  (12  +  15  +  20)  parts  altogether,  etc. 

Ex.  4.  Divide  £  11.  12s.  between  12  men,  8  women,  and  20  chil- 
dren, giving  to  each  man  twice  as  much  as  to  each  woman,  and 
to  each  woman  three  times  as  much  as  to  each  child. 

A  man's  share  =  a  woman's  share  x  2  =  a  child's  share  x  6. 
Hence  12  men,  8  women,  and  20  children  will  have 

(12  x  6  +  8  x  3  +  20)  shares  of  a  child. 
Hence  a  child's  share  x  (72  +  24  +  20)  =  £  11.  12s.  =  232s. ; 

.  •.  a  child's  share  =  f  f  |s.  =  2s. 
Whence  it  follows  that  each  man  has  12s.,  and  each  woman  6s. 


Art.  210.]  EXAMPLES.  219 

Ex.  5.  Divide  532  into  three  parts  proportional  to  f ,  f ,  and  $. 

Since  f ,  f ,  and  f  are  respectively  ff ,  ££,  and  £§,  we  have  merely 
to  divide  into  parts  proportional  to  40,  45,  and  48. 

40 

Hence,  as  in  Ex.  3,  the  parts  are  - of  the  whole,  etc. 

40  +  45  +  48 


EXAMPLES   LXXXVIII. 
Written  Exercises. 

1.  Divide  $245  into  parts  in  the  ratio  3  :  4. 

2.  Divide  $165  into  parts  in  the  ratio  2\  :  3. 

3.  Divide  $33.15  into  parts  in  the  ratio  §  :  f . 

4.  Divide  $54  into  three  parts  proportional  to  the 
numbers,  1,  2,  and  3. 

5.  Divide  $90.19  into  parts  proportional  to  the  num- 
bers, 7,  9,  and  13. 

6.  Divide  £17.  lis.  into  three  parts  proportional  to  5, 
51,  and  7f 

7.  A  sum  is  divided  into  parts  proportional  to  the 
fractions,  f ,  f ,  J ;  what  fractional  part  of  the  whole  is  the 
first  part  ? 

8.  The  profits  of  a  business  are  to  be  divided  between 
the  three  partners  in  proportion  to  the  numbers,  5,  3,  and 
2 ;  how  much  does  each  receive  out  of  a  total  profit  of 
$6237  ? 

9.  In  a  certain  business  A  has  7  shares,  B  5,  C  3,  and 
D  1  share.  The  profits  are  $2410.  Find  each  partner's 
share. 

10.  A  provides  $5000,  B  $3000,  and  C  $1250  to 
carry  on  a  business.  How  much  should  each  get  out  of  a 
profit  of  $555  ? 


220  RATIO  —  PROPORTION.  [Chap.  IX. 

11.  A,  B,  and  0  are  partners  in  a  business  and  have 
shares  in  proportion  to  the  numbers,  4,  3,  and  2,  respec- 
tively, after  ^  per  annum  has  been  paid  on  the  capital. 
The  capital  is  $20000,  of  which  sum  A  provided  $12000, 
and  B  the  remainder.  How  much  does  each  receive  out 
of  a  total  yearly  profit  of  $3400  ? 

12.  A,  B,  and  C  are  partners  in  a  business ;  C  as  mana- 
ger receives  y1^  of  the  net  profits,  the  remainder  being  di- 
vided between  A,  B,  and  C  in  proportion  to  the  numbers, 
5,  4,  and  3,  respectively.  In  a  certain  year  A's  share  of 
the  profits  amounted  to  $1520 ;  what  were  the  shares  of 
B  and  C  ? 

13.*   The  shares  of  A,  B,  and  C  of  the  capital  in  a 

business  are  as  4  to  3  to  2.     After  4  months  A  withdraws 

half  his  capital,  and  the  profits  at  the  end  of  the  year  are 

$1518.     How  should  this  be  divided  between  A,   B, 

andC? 

Hint.  A  has  {  4  f or  4  mos'  }  =  32  for  1  mo. 

(  2  for  8  mos.  > 

14.  Divide  $157.50  between  A,  B,  C,  and  D,  so  that 
A  may  have  as  much  as  C  and  D  together,  B  as  much  as 
A  and  C  together,  and  D  twice  as  much  as  C. 

15.  Divide  $80  among  22  men,  26  women,  and  82 
boys,  so  that  2  men  may  have  as  much  as  3  women,  and 
1  woman  as  much  as  2  boys. 

16.  If  8  men  can  do  as  much  as  14  women,  and  5 
women  as  much  as  9  boys,  divide  $270  among  4  men,  6 
women,  and  9  boys  in  proportion  to  the  work  they  do. 


*  When  partners  put  capital  into  a  business  for  the  same  length 
of  time,  the  case  is  one  of  Simple  Partnership. 

When  capital  is  put  into  a  business  for  different  lengths  of  time, 
the  case  is  one  of  Compound  Partnership. 


Arts.  211,212.]  MIXTURES.  221 

17.  Divide  $1519.10  among  three  persons,  A,  B,  and 
C,  so  that  A  may  get  one-fourth  as  much  as  B  receives, 
and  C  may  get  one-tenth  as  much  as  A  and  B  together. 

18.  Three  partners,  A,  B,  and  C,  had  shares  in  a  busi- 
ness proportional  to  the  numbers,  4,  5,  and  6,  respectively. 
C  retired  and  received  as  his  share  of  the  business  #15000. 
How  much  of  this  money  should  be  paid  by  A  and  B 
respectively  in  order  that  after  C's  retirement  their  shares 
might  be  equal  ? 

19.  A  and  B,  whose  capitals  were  as  3  to  4,  joined  in 
business,  and  at  the  end  of  4  months  they  withdrew  f 
and  f  respectively  of  their  capitals  from  the  business. 
How  should  a  gain  of  $624  be  divided  between  them  at 
the  end  of  the  year  ? 

20.  The  volumes  of  three  substances  contained  in  a 
certain  mixture  are  proportional  to  the  numbers,  2, 1,  and 
4,  respectively ;  also  the  weights  of  equal  volumes  of  the 
substances  are  as  the  numbers,  1,  32,  and  16,  respectively. 
Find  the  weight  of  the  first  substance  contained  in  3  lb. 
1  oz.  of  the  mixture. 

Mixtures. 

211.  The  cost  of  a  mixture  of  given  quantities  of  two 
different  ingredients  is  at  once  found  when  the  prices  of 
the  separate  ingredients  are  known. 

Ex.    8  lb.  of  tea  costing  30  ct.  per  pound  is  mixed  with  3  lb.  of 
tea  costing  55  ct.  per  pound;  ichat  is  the  cost  of  the  mixture? 
The  mixture  cost  30  ct.  x  8  +  55  ct.  x  3  =  405  ct. 
Hence,  each  pound  of  the  mixture  cost  405  ct.  -f-  11  =  36T9T  ct. 

212.  The  ratio  in  which  two  different  ingredients  must 
be  taken  in  order  to  make  a  mixture  whose  cost  is  any 
given  sum  intermediate  between  the  costs  of  the  separate 
ingredients,  will  be  seen  from  the  following  examples. 


222  RATIO  — PROPORTION.  [Chap.  IX. 

Ex.  1.  In  what  ratio  must  tea  costing  30  ct.  per  lb.  be  mixed  with 
tea  costing  55  ct.  per  lb.  that  the  mixture  may  cost  45  ct.  per  lb.  ? 

The  loss  on  the  better  quality  is  10  ct.  per  lb. 
The  gain  on  the  poorer  quality  is  15  ct.  per  lb. 
The  ratio  between  the  loss  and  gain  being  |,  we  equalize  loss 
and  gain  by  making 

the  number  of  lb.  of  the  better  quality  _  3 
the  number  of  lb.  of  the  poorer  quality     2 

Ex.  2.  In  what  way  must  3  kinds  of  tea  worth  30 ct.,  35 ct.,  and 
50  ct.  per  lb.  respectively,  be  mixed  that  the  mixture  may  be  worth 
38  ct.  per  lb.? 

When  there  are  3  (or  more)  kinds  of  commodity,  and  only  the 
price  of  the  mixture  fixed,  there  is  an  indefinite  number  of  ways 
of  satisfying  the  condition. 

In  the  present  case  the  gain  on  the  lower  two  grades  of  tea, 
namely,  11  ct.  on  2  lb.  (1  lb.  of  each  grade)  must  just  balance 
the  loss  on  the  best  grade,  namely,  12  ct.  per  lb.  The  ratio  be- 
tween gain  and  loss  =  \\.  Hence,  we  must  have  12  lb.  of  each  of 
the  lower  grades  and  11  lb.  of  the  best  grade. 

Or,  we  may  say  that  the  gain  on  2  lb.  of  30  ct.  tea  with  the  gain 
on  1  lb.  of  35  ct.  tea  (19  ct.  in  all)  must  just  balance  the  loss  (24  ct.) 
on  a  certain  number  of  2  lb.  packages  of  50  ct.  tea.  Here  the  ratio 
of  gain  to  loss  is  |$,  Hence,  we  must  have  twenty-four  3  lb. 
packages  (each  package  consisting  of  2  lb.  of  30  ct.  tea  and  1  lb.  of 
35  ct.  tea)  and  nineteen  2  lb.  packages  of  the  50  ct.  tea. 

Ans.  =48 lb.,  241b.,  and  381b. 

Or,  gainon{llb-30ct-tea=    8cH  =  20ct.; 

1 4  lb.  35  ct.  tea  =  12  ct.  J 

loss  on      4  lb.  50  ct.  tea  =  48  ct. ; 

.-.  gain  :  loss  : :  5  :  12. 

Hence,  we  must  have  twelve  5  lb.  packages  (1  lb.  of  first  kind 
with  4  lb.  of  second  kind)  and  five  4  lb.  packages  of  third  kind. 

Ans.  =  12  lb.,  48  lb.,  and  20  lb. 


Art.  213.]  WORK  AND  TIME.  223 

EXAMPLES  LXXXIX. 
Written  Exercises. 

1.  What  would  be  the  cost  per  lb.  of  a  mixture  of 
4  lb.  of  tea  at  30  ct.,  and  6  lb.  at  40  ct.  ? 

2.  What  will  be  the  cost  of  a  mixture  of  3  gal.  of  spirit 
at  $2.80  per  gal.  and  5  gal.  at  $3.50  a  gal.  ? 

3.  If  180  lb.  of  sugar  which  cost  4  ct.  per  lb.  be  mixed 
with  120  lb.  which  cost  5¥  ct.  per  lb.,  at  what  price  must 
the  mixture  be  sold  so  as  to  gain  let.  per  lb. 

4.  A  milkman  buys  milk  at  20  ct.  per  gal.  He  adds 
£  as  much  water  as  he  buys  milk,  and  sells  the  mixture 
at  28  ct.  per  gal.     What  is  his  gain  per  gal.  ? 

5.  In  what  ratio  must  two  kinds  of  tea,  which  cost 
respectively  Is.  3d.  and  Is.  9&  per  pound,  be  mixed  in 
order  that  the  mixture  may  cost  Is.  5d.  per  pound  ? 

6.  In  what  ratio  must  biscuits  worth  respectively 
11  ct.  per  lb.  and  15  ct.  per  lb.  be  mixed  that  the  mixture 
may  be  worth  12  ct.  per  lb.  ? 

7.  How  much  sugar  worth  7Jct.  per  lb.  must  be 
mixed  with  112  lb.  of  sugar  worth  4£  ct.  per  lb.  in 
order  that  the  mixture  may  be  worth  7  ct.  per  lb.  ? 

8.  Tea  at  66  ct.  a  lb.  is  mixed  with  tea  at  78  ct.  a 
lb.  In  what  proportion  must  they  be  mixed,  so  that 
by  selling  the  mixture  at  77  ct.  a  lb.  a  profit  of  y1^  of 
the  cost  may  be  made  ? 

Work  and  Time. 

213.  We  now  consider  problems  with  reference  to  work 
done  in  various  times.  These  can  all  be  solved  by  con- 
sidering the  fractional  parts  of  the  whole  work  which  are 
done  in  a  definite  time. 


224  RATIO  — PROPORTION.  [Chap.  IX. 

Ex.  1.  One  man  can  mow  afield  in  30  hr.,  and  another  man 
can  mow  the  field  in  60  hr.  ;  how  long  would  it  take  them  working 
together  to  do  it  f 

The  first  man  mows  fa      of  the  whole  in  1  hr., 

the  second  man  mows  -X      of  the  whole  in  1  hr. ; 


And,  as  the  two  together  would  mow  (fa  +  fa)  =  fa  of  the 
whole  in  1  hr.,  they  would  mow  the  whole  in  20  hr. 

Ex.  2.  A  cistern  could  be  filled  in  20  min.  by  its  supply  pipe 
and  emptied  in  35  min.  by  its  waste  pipe.  If  the  cistern  be  empty 
and  both  pipes  be  opened,  how  long  would  it  take  to  fill  it  ? 

The  supply  pipe  fills  fa  of  the  cistern  in  1  min., 
the  waste  pipe  empties  fa  of  the  cistern  in  1  min. ; 
hence,  together  they  fill  (fa  —  fa)  =  T|¥  of  the  cistern  in  1  min. 

And,  as  Tf ^  of  the  whole  is  filled  in  1  min.,  the  whole  will  be 
filled  in  1  min.  -?-  Tf7  =  46 1  min. 


EXAMPLES   XC. 
Written  Exercises. 

1.  A  can  mow  a  field  in  3  da.,  and  B  can  mow  the 
same  field  in  6  da. ;  in  how  many  da.  will  they  do  it 
working  together  ? 

2.  A  bath  could  be  filled  by  its  cold  water  pipe  in  15 
min.  and  by  its  hot  water  pipe  in  30  min. ;  in  what  time 
will  it  be  filled  when  both  are  opened  ? 

3.  A  can  do  a  piece  of  work  in  12  da.,  and  B  can  do 
the  same  in  20  da.  A  works  at  it  for  3  da.  How  long 
would  it  take  B  to  finish  it  ? 

4.  A  can  mow  a  field  in  15  hr.,  and  B  can  mow  the 
same  field  in  25  hr.  They  work  together  for  1\  hr., 
when  A  goes  away.  How  long  will  it  take  B  to  finish 
the  work  ? 


Art.  213.]  WORK  AND  TIME.  225 

5.  Two  men  together  can  do  in  20  days  a  piece  of 
work  which  one  of  them  alone  could  do  in  30  days ;  how 
long  would  it  take  the  other  man  to  do  the  work  alone  ? 

6.  When  the  hot  and  cold  water  pipes  are  both 
opened  a  bath  is  filled  in  6  minutes ;  and  when  only  the 
cold  water  is  turned  on,  the  bath  is  filled  in  10  minutes. 
In  how  long  would  the  bath  be  filled  if  the  hot  water 
pipe  only  were  opened  ? 

7.  A  and  B  could  together  finish  a  piece  of  work  in  25 
days.  They  work  together  for  15  days,  and  then  A 
finished  it  by  himself  in  20  days.  How  long  would  it 
take  them  to  do  the  whole,  working  separately  ? 

8.  A  and  B  could  together  do  a  piece  in  22^-  days. 
A  worked  at  it  alone  for  10  days,  and  then  B  finished  it 
alone  in  60  days.  How  long  would  it  take  them  sepa- 
rately to  do  the  whole  work  ? 

9.  A  can  do  a  piece  of  work  in  2\  days,  B  can  do  it 
in  3  days,  and  C  can  do  it  in  3f  days ;  how  long  would  it 
take  them  to  do  it,  all  working  together  ? 

10.  A  cistern  is  filled  by  one  pipe  in  48  minutes,  by 
another  in  an  hour,  and  by  a  third  in  half  an  hour ;  in 
what  time  would  it  be  filled  if  all  three  pipes  were  open 
together  ? 

11.  A  cistern  can  be  filled  by  one  pipe  in  3  hours,  by 
another  in  3  hr.  and  40  min.,  and  it  can  be  emptied  by  a 
third  pipe  in  2  hr.  20  min. ;  if  it  be  empty,  and  they  are 
all  opened  together,  in  what  time  will  the  cistern  be 
filled  ? 

12.  C  does  half  as  much  in  a  day  as  A  and  B  can  do 
together,  and  B  does  half  as  much  again  as  A ;  if  all 
three  working  together  can  mow  20  acres  of  barley  in  16 
days,  how  long  would  each,  working  by  himself,  take  to 
mow  5  acres  ? 


226  RATIO  — PROPORTION.  [Chap.  IX. 

13,  A  can  do  a  piece  of  work  in  6  days,  B  in  8  days, 
and  C  in  12  days.  B  and  C  work  together  for  2  days,  and 
then  C  is  replaced  by  A.  Find  when  the  work  will  be 
finished. 

14.  A  and  B  together  can  perform  a  piece  of  work  in 
24  hr.,  A  and  C  in  30  hr.,  and  B  and  C  in  40  hr. ;  in 
what  time  would  each  be  able  to  perform  it  when  work- 
ing separately  ? 

Races  and  Games. 

214.  The  following  are  examples  of  questions  of  this 
nature. 

Ex.  1.  In  a  100  yards  race  A  can  give  B  5  yards  start  and  just 
win  ;  also,  B  can  give  C  5  yards  start ;  how  much  could  A  give  C  ? 

A  runs  100  yards  while  B  runs  95,  and  B  runs  100  yards  while 
C  runs  95. 

Hence,  C's  distance  in  any  time  =  T9^  of  B's  =  fifo  x  T9o5^  of  A's. 

Hence,  while  A  runs  100  yards,  C  will  run  T9^  x  TV<j-  of  100  =  90£ 
yards. 

Thus,  A  can  give  9|  yards  to  C. 

Ex.  2.  In  a  certain  game  A  can  give  B  1  point  in  5,  B  can  give 
C  1  point  in  5,  and  G  can  give  D  1  point  in  8 ;  how  many  points 
in  100  can  A  give  D  t 

A  :  B  =5:4  =  25  :  20 

B :  C         =        5:4        =         20 :  16 

C  :  D  =  8:7=  16  :  14. 

Hence,  as  in  Ex.  3,  Art.  210, 

A  :  B  :  C  :  D  : :    25  :  20  :  16  :  14 

: :  100  :  80  :  64  :  56. 

Thus,  A  can  give  (100  -  56  =)  44  points  in  100  to  D ;  i.e.,  A  can 
make  100  points  while  D  makes  56  points. 


Art.  214.]  EXAMPLES.  227 

EXAMPLES  XCI. 
Written  Exercises. 

1.  A  can  give  B  10  yards  start  in  a  race  of  100  yards, 
and  B  can  give  C  10  yards  start  over  the  same  distance. 
How  many  yards  start  can  A  give  C  ? 

2.  A  can  give  B  20  yards  and  C  51  yards  start  in  a 
quarter  of  a  mile  race.  How  many  yards  could  B  give 
C  in  a  quarter  of  a  mile  ? 

3.  A  can  beat  B  by  5  yards  in  a  100  yards  race,  and 
B  can  beat  C  by  10  yards  in  a  200  yards  race ;  by  how 
much  could  A  beat  C  in  a  400  yards  race,  supposing  that 
they  always  run  at  the  same  pace  ? 

4.  A  wins  a  race  of  100  yards,  beating  B  by  19  yards 
and  C  by  10  yards ;  how  many  yards  start  ought  C  to 
give  B  in  200  yards  that  they  may  run  a  dead  heat  ? 

5.  In  a  certain  game  A  can  give  B  1  point  in  10,  B 
can  give  C  1  point  in  6 ;  how  many  can  A  give  C  in  100  ? 

6.  At  a  certain  game  A  scores  100  points  while  B 
scores  85,  and  B  scores  100  while  C  scores  80;  how 
many  will  C  score  in  the  time  that  it  takes  A  to  score  500  ? 

7.  A  can  make  9  articles  while  B  makes  14,  and  B 
can  make  7  while  C  makes  6 ;  how  many  can  C  make  in 
the  time  that  A  makes  30  ? 

8.  In  a  certain  game  A  can  give  B  1  point  in  10,  B 
can  give  C  1  point  in  12,  and  C  can  give  D  1  point  in  15 ; 
how  many  can  A  give  D  in  1000  ? 

9.  A  can  give  B  20  yards  and  can  give  C  41  yards 
start  in  a  race  of  a  quarter  of  a  mile,  and  B  can  give  C  a 
start  of  3  seconds  over  the  same  distance;  how  long 
does  each  take  to  run  a  quarter  of  a  mile  ? 


228  RATIO  — PROPORTION.  [Chap.  IX. 

10.  In  a  certain  game  A  can  give  B  1  point  in  5,  B  can 
give  C  1  point  in  8,  and  C  can  give  D  3  points  in  10 ; 
how  many  can  A  give  D  in  100  ? 

215.   The  following  examples  are  worth  notice. 

Ex.  1.  A  starts  at  10  o'clock  to  walk  along  a  road  at  the  rate  of 

4  miles  an  hour  ;  B  starts  on  a  tricycle  at  45  minutes  past  10  and 
rides  after  A  at  the  rate  of  9  miles  an  hour.  When  will  B  over- 
take A  ? 

When  B  starts,  A  has  already  traveled  £$  of  4  miles  ;  that  is,  3 
miles. 

B  gains  on  A  at  the  rate  of  (9  —  4  =  )  5  miles  an  hour. 

B  will  overtake  A  when  he  has  gained  3  miles,  which  he  will  do 
in  (3  -r-  5)  hours  =  36  minutes. 

Ex.  2.  At  what  time  between  4  and  5  o'clock  will  the  hands  of  the 
clock  be  together  ? 

At  4  o'clock  the  minute-hand  is  20  minute-spaces  behind  the 
hour-hand.  In  one  hour  the  minute-hand  passes  over  60  minute- 
spaces,  and  the  hour-hand  passes  over  5  minute-spaces. 

Thus,  the  minute-hand  gains  55  minute-spaces  in  an  hour. 

Now,  when  the  two  hands  are  together,  the  minute-hand  must 
have  gained  on  the  hour-hand  20  minute-spaces,  and  the  time 
required  for  this  =  §£  of  an  hour  =  2lT9T  minutes. 

Thus,  the  time  required  is  21T9T  minutes  past  4. 

Ex.  3.  A  train  traveling  at  the  rate  of  45  miles  an  hour  is 
observed  to  completely  pass  a  certain  telegraph  post  in  5  seconds  ; 
it  also  completely  passed  in  4  seconds  a  second  train  which  was 
traveling  along  a  parallel  line  of  rails  in  the  opposite  direction  at 
the  rate  of  30  miles  an  hour.     How  long  were  the  trains  ? 

The  time  the  first  train  takes  to  completely  pass  a  post  is  the 
time  it  takes  to  travel  a  distance  equal  to  the  length  of  the  train  ; 
and,  since  the  train  goes  at  the  rate  of  45  miles  an  hour,  it  goes  in 

5  seconds  a  distance  =  jf^  of  45  miles  =  110  yards.  Thus,  the 
first  train  is  110  yards  long. 

Again,  in  the  time  the  trains  take  to  completely  pass  one  another 
the  distance  traveled  by  the  two  trains  together  must  be  the  sum 


Art.  215.]  EXAMPLES.  229 

of  the   lengths  of   the  trains;    and  in  4  seconds  the  trains  will 
together  travel  ^%7  of  45  miles  +  ¥ ^  of  30  miles  =  146  yd.  2  ft. 
Hence  the  length  of  the  second  train 

=  146  yd.  2  ft.  -  110  yd.  =  36  yd.  2  ft. 

Ex.  4.  Seven  fowls  are  worth  6  ducks,  7  ducks  are  icorth  2  geese, 
10  geese  are  worth  7  turkeys,  and  a  turkey  is  icorth  17s.  6d.  ;  how 
much  is  a  fowl  worth  ? 


"  "  f  of  f  of  the  worth  of  a  goose, 

"  "         T\-  of  f  of  f  of  the  worth  of  a  turkey, 

"  "         ^of  f  of  f  of  17s.  6c?. 

=  TV  x  f  x  f  x  ^  shillings  =  3s. 

EXAMPLES   XCII. 
"Written  Exercises. 

1.  One  boy  runs  at  the  rate  of  100  yards  in  15  sec- 
onds, and  has  a  start  of  40  yards  in  front  of  another  boy 
who  runs  at  the  rate  of  100  yards  in  12  seconds ;  when 
will  the  first  boy  be  overtaken  ? 

2.  One  cyclist  rides  at  the  rate  of  15  miles  an  hour 
and  starts  half-an-hour  after  another  who  rides  along  the 
same  road  at  the  rate  of  12  miles  an  hour;  when  will 
the  first  rider  be  overtaken  ? 

3.  At  what  time  between  5  and  6  o'clock  will  the 
hands  of  a  clock  be  together  ? 

4.  At  what  time  between  2  and  3  o'clock  will  the 
hands  of  a  clock  be  at  right  angles  ? 

5.  A  train  traveling  at  the  rate  of  45  miles  an  hour 
is  observed  to  completely  pass  a  certain  point  in  9  seconds ; 
find  the  length  of  the  train. 


230  RATJO  —  PROPORTION.       [Chaps.  IX.,  X. 

6.  A  man  on  the  platform  of  a  station  observed  that 
a  train  passed  him  in  10  seconds,  and  passed  completely 
through  the  station,  which  is  308  yards  long,  in  24  seconds ; 
how  long  was  the  train,  and  how  fast  was  it  going  ? 

7.  A  passenger  train,  moving  at  the  rate  of  45  miles 
an  hour,  overtook  a  mineral  train  twice  as  long  as  itself 
and  which  was  going  along  a  parallel  line  of  rails  in  the 
same  direction  at  the  rate  of  23  miles  an  hour ;  and  the 
passenger  train  completely  passed  the  mineral  train  in 
22^-  seconds.     How  long  was  each  train  ? 

8.  A  person  lights  two  candles,  12  and  10  inches  long 
respectively,  at  6  p.m.  The  former  diminishes  5  inches 
in  length  in  4  hours,  and  the  latter  1  inch  in  2  hours. 
If  kept  alight,  at  what  time  will  the  former  be  two  inches 
shorter  than  the  latter  ? 

9.  If  3  pears  are  worth  as  much  as  4  apples,  5  apples 
as  much  as  3  plums,  8  plums  as  much  as  3  peaches,  and 
if  pears  cost  36  ct.  a  dozen,  what  is  the  price  of  a  peach  ? 

10.  Twelve  fowls  are  worth  as  much  as  11  ducks,  5 
ducks  are  worth  as  much  as  4  pheasants,  10  pheasants  as 
much  as  3  turkeys,  and  7  turkeys  as  much  as  10  geese ; 
also  a  fowl  and  a  pheasant  are  together  worth  6s.  6d. 
Find  the  cost  of  a  goose  and  a  turkey  together. 


Arts.  216,217.]       PERCENTAGE   A   PATIO.  231 


CHAPTER   X. 

PERCENTAGES. 
Percentage  a  Ratio. 

216.  In  many  cases  the  ratio  of  one  number  to  another, 
or  of  one  quantity  to  another  of  the  same  kind,  is  ex- 
pressed by  the  number  of  hundredths  the  first  is  of  the 
second,  and  this  is  called  the  per  cent  the  first  is  of  the 
second. 

For  example,  2  apples  =  T2^  of  8  apples,  or  25  per  cent  of  8 
apples. 

This  means  that  the  ratio  of  2  apples  :  8  apples  is  .25.  The  first 
term  is  sometimes  called  the  percentage,  the  second  is  called  the 
base,  and  the  quotient  is,  as  formerly,  the  ratio. 

Per  cent  is  expressed  by  the  sign  %,  or  by  writing  the  numerator 
as  a  decimal ;  thus,  T2^  =  25%,  or  .25  ;  and  we  write  '  5  is  25%  of 
20,'  or  '5  is  .25  of  20.' 

217.  The  following  examples  will  show  how  to  ex- 
press any  given  quantity  as  a  per  cent  of  any  other  given 
quantity  of  the  same  kind. 

Ex.  1.   Five  dollars  is  what  %  o/$40  ? 

$5  =  &  of  $40  =  i  of  $40  =  i?i  of  $40  =12  \  %  of  $40. 

Ex.  2.  In  a  town  whose  population  was  243200  there  were  15504 
children  born  in  a  year.  Find  the  per  cent  the  number  of  births  was 
of  the  population. 

The  ratio  of  births  to  population  is  15504  :  243200  ; 
15504  +  243200  =  .06| ;  .-.  Ans.  =  6j  %. 


232  PERCENTAGES.  [Chap.  X. 

218.  The  following  examples  will  show  how  to  find  a 
given  per  cent  of  a  given  quantity. 

Ex.  1.   Find  12£%  of  $18. 

12|%  of  $18  =  .12|  of  #18  =  $2.25. 

Ex.  2.    In  a  town  whose  population  was  243200  the  birth  rate  in  a 

year  was  6|%  of  the  population  ;  how  many  children  were  born  in 

the  year. 

6f  %  of  243200  =  .06|  times  243200  =  15504. 

Ex.  3.    Of  what  is  9  ct.  22 \%  ? 

Since  9  ct.  is  22 \%,  100%,  or  the  whole,  must  be  ~5  of  9ct. 
=  -V-  of  9  ct.  =  40  ct.  22* 

219.  Frequently  in  finding  percentage  it  is  best  to 
multiply  by  the  common  fraction  which  is  equivalent  to 
the  per  cent  expressed  decimally ;  thus, 

6\%  of  96  =  T\  of  96  ;  12 l%  of  432  =  |  of  432  ; 

16f  %  of  $36.85  =  i  of  $36.85  =  $6.14^.      [Art.  134.] 

EXAMPLES  XCIII. 
Oral  Exercises. 

What  fractions  are  denoted  by  the  following  per  cents  ? 

1.  50.  4.   10.  7.   12J.  10.    3^. 

2.  25.  5.    5.  8.    16}.  11.    6J, 

3.  20.  6.    2\.  9.    33|.  12.    5|. 

What  per  cents  are  equivalent  to  the  following  fractions? 

13.  \.  16.    \.  19.    A.  22.    TV 

14.  f.  17.    TV  20.    \\.  23.    &. 

15.  f.  18.    fr  21.    ^.  24.    if 

Written  Exercises. 

Find  the  per  cent  the  first  is  of  the  second  in 

25.  $10,  $25;  45  ct,  $2.70;  $1,121  $3. 

26.  7.s.  6d,  £2.  27.    7  lb.,  1  cwt. 


Arts.  218,219.] 


EXAMPLES. 


233 


28.  1  hr.  12  min.,  1  da. 

29.  3oz.  15dwt.,  lib.  lOdwfc 

30.  3216 %  1.608 * 
Find 

31.  5%  of  £7.  10s. 

32.  10%  of  $85.63, 

33.  12|%  of  $492.64. 

34.  The  population  of  a  certain  town  increased  50  % 
in  the  10  years  from  1881  to  1891,  and  the  population 
in  1891  was  34617 ;  what  was  the  population  in  188.1  ? 

35.  Find  the  %  of  error  in  the  statement  that  1  oz. 
Troy  is  equal  to  1.1  oz.  Avoir. 

36.  Fill  the  blanks  in  the  following  table  by  giving 
the  per  cents  of  the  1889  amounts  to  the  nearest  tenth. 


Receipts  from  — 

1890. 

16S9. 

Increase. 

Amount. 

Per 
Cent. 

Ordinary  passengers    .  . 
Season-ticket  holders  .  . 
Excess  baggage,  mails,  etc. 

Total 

26983000 
2316000 
5029000 

25678000 
2196000 
4757000 

1305000 
120000 
272000 

34328000 

32631000 

1697000 

37.   Fill  the  blanks  in  the  following  table  by  giving 
the  per  cents  of  the  1889  amounts  to  the  nearest  tenth. 


Receipts  from  — 

1S90. 

18S9. 

Increase. 

Amount. 

Per 

Cent. 

Mineral  traffic 

General  mdse  traffic    .  . 
Live  stock 

Total 

17543000 

23300000 

1377000 

$ 

17052000 

22694000 

1340000 

1 

491000 

606000 

37000 

42220000 

41086000 

1134000 

234  PERCENTAGES.  [Chap.  X. 


Profit  and  Loss. 

220.  When  anything  is  sold  for  more  than  it  cost,  it  is 
said  to  be  sold  at  a  profit,  and  when  it  is  sold  for  less 
than  it  cost,  it  is  said  to  be  sold  at  a  loss.  Profit  or  loss 
is  often  expressed  as  a,  percentage,  and  this  percentage  is 
always  to  be  reckoned  on  the  cost  price. 

Thus,  if  goods  which  cost  $50  are  sold  for  $60,  the  percentage 
gain,  or  profit,  is  $10,  and  the  per  cent  gain  is  10 :  50  =  20  %  on 
the  original  outlay. 

221.  The  following  examples  will  show  how  to  treat 
questions  involving  profit  or  loss. 

Ex.  1.  A  house  was  bought  for  $400  and  sold  for  $480 ;  what 
was  the  profit  per  cent  ? 

The  total  profit  =  $480  -  $400  =  $80. 
And  the  ratio  of  $80 :  $400  =  20%. 

Ex.  2.  An  article  cost  $10.40  and  was  sold  at  a  loss  of  15% ;  for 
what  was  it  sold  t 

Selling  price  =  cost  —  15%  of  cost ; 
.-.       "         "     =  85%  of  cost  =  .85  of  $10.40  =  $8.84. 

Ex.  3.  What  was  the  cost  of  goods  which  were  sold  for  $56,  at  a 
gain  of  12%? 

Selling  price  =  cost  +  12%  of  cost  =  112%  of  cost ; 
.♦.  cost  =  i£f  of  selling  price  =  $50. 

Ex.  4.  By  selling  tea  at  50  ct.  a  pound  a  grocer  would  gain  5% 
more  than  by  selling  it  at  48  ct.  a  pound  ;  what  was  the  cost  of  the 
tea  ? 

50  ct.  -  48  ct.  is  5%  of  the  cost ;  hence  2  ct.  is  5%  of  the  cost ; 
.-.  40  ct.  =  the  cost. 


Arts.  220, 221.]  PROFIT  AND  LOSS.      .  235 

Ex.  5.  A  manufacturer  sells  at  a  profit  of  20%  to  a  wholesale 
dealer,  who  sells  at  a  profit  of  15%  to  a  retail  dealer,  and  the  retail 
dealer  sells  for  $2.76  and  makes  a  profit  of  25%.  Find  the  cost 
of  manufacture. 

It  cost  the  retail  dealer  {%$  of  $2.76 

"       "   wholesale  dealer        fff  of  }f§  of  $2.76 

"       "   manufacturer  ff  §  of  if  §  of  iff  of  $2.76 

Thus  required  cost  =  {§-<>-  x  |fo  x  i  go  of  $2.76  =  $1.60. 

EXAMPLES    XCIV. 
Written  Exercises. 

What  was  the  gain  or  loss  %  in  the  following  cases  ? 

1.  Cost  price  $20,  selling  price  $ 24. 

2.  Cost  price  $2.00,  selling  price  $2.28. 

3.  Cost  price  40  ct.,  selling  price  44  ct. 

4.  Cost  price  $3,  selling  price  $3.60. 

5.  Cost  price  $140,  selling  price  $130. 

6.  Cost  price  $1.20,  selling  price  $1.62. 

7.  Cost  price  84  ct.,  selling  price  98  ct. 

8.  Cost  price  $7.80,  selling  price  $8.97. 

9.  Cost  price  $74,  selling  price  $70.30. 

10.  Cost  price  $15.20,  selling  price  $20.52. 

11.  Cost  price  $12.40,  selling  price  $10.23. 

12.  Cost  price  $147,  selling  price  $122.01. 

13.  If  an  article  be  bought  for  $4.20  and  sold  for 
$6.60,  what  is  the  gain  %  ? 

14.  What  was  the  cost  price  of  tea  which  is  sold  for 
80  ct.  a  pound  and  at  a  gain  of  25%  ? 

15.  If  a  grocer  buys  60  lb.  of  tea  for  $21.00,  at  what 
price  per  lb.  must  he  sell  it  so  as  to  make  20%  profit? 


236  PERCENTAGES.  [Chap.  X. 

16.  An  article  was  sold  for  56  ct.,  at  a  gain  of  12%  ; 
what  did  it  cost  ? 

17.  The  profit  on  an  article  if  sold  for  $3.00  is  25% ; 
what  would  be  the  profit  if  it  were  sold  for  $2.88  ? 

18.  By  selling  a  house  for  $759  a  builder  gained  10%  ; 
what  would  he  have  lost  %  if  he  had  sold  for  $621  ? 

19.  If  a  profit  of  22-1-%  is  made  by  selling  an  article 
for  $2.94,  what  would  be  the  selling  price  if  the  profit 
were  only  5%  ? 

20.  A  person  bought  a  carriage  and  sold  it  for  $37.80 
more  than  he  gave  for  it,  thereby  clearing  7%  ;  what 
did  he  give  for  it  ? 

21.  A  house  is  sold  for  $4000,  and  25%  profit  is  made ; 
how  much  %  profit  would  be  made  by  selling  for  $3360  ? 

22.  A  tradesman  by  selling  an  article  for  $1.62  gains 
35%  ;  what  would  he  have  gained  %  if  he  had  sold  it 
for  $1.98  ? 

23.  A  man  bought  apples  at  the  rate  of  6  for  2  ct., 
and  an  equal  number  at  the  rate  of  10  for  2  ct. ;  and  he 
sold  the  whole  at  the  rate  of  5  for  2  ct.  What  profit  % 
did  he  make  ? 

24.  If  5%  more  be  gained  by  selling  an  article  for  24 
ct.  than  by  selling  it  for  23  ct.,  what  was  the  original 
price  ? 

25.  If  3%  more  be  gained  by  selling  a  horse  for 
$399.60  than  by  selling  for  $388.80,  what  must  have 
been  the  original  cost  ? 

26.  If  a  woman  gains  12%  by  selling  5  herrings  for 
14  ct.,  what  %  would  she  gain  by  selling  them  at  6  for 
18  ct.? 

27.  If  a  woman  buys  eggs  at  20  ct.  a  dozen,  how  many 
ought  she  to  sell  for  18  ct.  in  order  to  gain  8%  ? 


Art.  222.]  TRADE   DISCOUNT.  237 

28.  A  man  who  had  been  paying  $25.20  for  4t.  of 
coal  changed  his  coal  merchant  and  then  got  5 1.  for 
$20.16 ;  how  much  did  he  save  %  ? 

29.  A  draper  bought  240  yd.  of  silk.  He  sold  J  at  a 
gain  of  25%,  J  at  a  gain  of  20%,  and  the  remainder  at 
a  loss  of  15%,  and  received  $800  in  all.  What  was  the 
cost  price  per  yd.  ? 

30.  A  draper  bought  a  piece  of  silk  35  yd.  long ;  and, 
after  cutting  off  2  yd.  which  were  damaged,  he  sold  the 
remainder  so  as  to  clear  10%  on  his  outlay.  How  much 
%  was  the  selling  price  of  a  yd.  higher  than  the  cost 
price  ? 

31.  A  manufacturer  sold  at  a  profit  of  25%  to  a 
wholesale  dealer,  who  sold  at  a  profit  of  12%  to  a  retail 
dealer,  and  the  retail  dealer  sold  for  $3.22  and  made  a 
profit  of  15%  ;  what  was  the  cost  of  manufacture  ? 

32.  A  quantity  of  wheat  was  sold  in  succession  by 
three  dealers,  each  of  whom  made  a  profit  of  5  % .  The 
last  of  the  three  sold  for  $3087 ;  how  much  did  it  cost 
the  first? 

33.  A  house  was  sold  by  the  builder  at  a  profit  of  30%, 
and  the  purchaser  sold  it  again  at  an  advance  of  $117  in 
the  price,  and  gained  20%  on  his  outlay;  how  much 
did  the  house  cost  the  builder  ? 


Trade  Discount. 

222.  Merchants  often  sell  goods  at  a  certain  price  with 
a  certain  %  discount ;  thus, 

Macmillan  &  Co.  may  sell  books  at  $1.60  per  copy  less 
15%  ;  this  means  that  they  sell  for  $1.60  -  15%  of  $1.60, 
or  for  $1.60  -  $.24  =  $1.36. 


238  PERCENTAGES.  [Chap.  X. 

223.  Sometimes  after  a  given  %  discount  is  allowed,  a 
second  allowance  of  another  %  is  made,  and  even  a  third 
allowance  is  made. 

Ex.  1.  Goods  sold  for  $2500  with  a  discount  of  20%,  5%,  and 
1 1  %  bring  what  price  f 

$2500  -  20  %  of  $2500  =  $2000  ; 
$2000  -  5  %  of  $2000  =  $1900  ; 
$1900  -  I*  %  of  $1900  =  $1871.50  =  Ans. 

Ex.  2.  Which  is  cheaper,  to  buy  goods  at  a  discount  of  30%  and 
5%,  or  with  $&\%off? 

The  marked  price  less  30%  =  70%  of  marked  price  ;  70%  -5% 
of  70  %  =  m\  %.  It  is  cheaper  to  buy  at  a  discount  of  30  %  and  5  % 
than  at  a  discount  of  33^  %. 

EXAMPLES  XCV. 
Oral  Exercises. 

What  is  paid  for  goods  marked 

1.  $50  with  a  discount  of  10%  ? 

2.  $50  with  a  discount  of  10%  and  10%  ? 

3.  $50  with  a  discount  of  20%  and  5%  ? 

4.  $600  with  a  discount  of  33|%  ? 

5.  $900  with  a  discount  of  16|%  ? 

6.  $1000  with  a  discount  of  27%  and  10%  ? 

7.  $1000  with  a  discount  of  20%,  10%,  and  1%  ? 

What  is  the  marked  price  of  goods  sold  for 

8.  $90  after  a  discount  of  25%  ? 

9.  $63  after  a  discount  of  30%  and  10%  ? 

10.  $49  after  a  discount  of  121%,  and  12  J  %  ? 

11.  $45  after  a  discount  of  16} %  and  10%  ? 


Arts.  223,  224.]     COMMISSION  — BROKERAGE.  239 

Written  Exercises. 

12.  Find  what  was  received  for  goods  marked  $1200 
if  a  discount  of  \  and  15%  is  allowed. 

13.  For  what  %  of  the  marking  price  are  goods  sold 
if  an  allowance  of  \,  10%,  and  6|%  is  made  ? 

14.  Goods  are  marked  $170  and  sold  for  $144.50; 
what  %  discount  was  allowed  ? 

15.  Goods  marked  $16  were  sold  at  6J%  discount  and 
5%  off  for  cash  ;  what  was  the  selling  price  ? 

16.  Goods  cost  a  merchant  $1600;  he  wishes  to  make 
a  profit  of  25%  after  making  a  discount  of  20%  and  16|%  ; 
what  was  the  marked  price  '? 

17.  At  what  %  above  the  cost  must  goods  be  listed 
that  a  merchant  may  allow  a  discount  of  20%  and  realize 
a  profit  of  12%  ? 

18.  A  merchant  allows  on  $2000  worth  of  goods  (list 
price)  a  discount  of  15%,  9%,  and  5%  for  cash,  then  |% 
to  clinch  the  bargain;  how  much,  cash  did  he  receive  and 
what  profit  did  he  make,  his  %  of  profit  being  8  ? 

Commission  and  Brokerage. 

224.  An  agent  employed  to  buy,  or  sell,  goods,  or  to 
collect  rents,  is  usually  paid  a  percentage  on  the  price  of 
the  goods,  or  on  the  amount  of  rent.  This  percentage  is 
called  Commission. 

To  insure  against  loss  of  life,  or  damage  by  fire,  some 
persons  pay  money  to  an  Insurance  Company.  In  return 
for  this  money,  the  Company  undertake  to  compensate 
the  person  insured  for  any  loss  caused  by  fire,  or  to  pay 
a  specified  sum  to  relatives  of  the  deceased.     The  money 


240  PERCENTAGES.  [Chap.   X. 

paid  to  the  Company  is  a  percentage  on  the  value  of  the 
property  insured,  or  on  the  specified  sum,  and  is  called  a 
Premium. 

Kx.  1.  The  total  rental  of  an  estate  is  $8474.40,  and  the  agent 
is  paid  a  commission  of  5%;    how  much  is  the  commission/ 

$8474.40  x  .05  =  $423.72. 

Ex.  2.   What  is  the  annual  premium  for  insurance  on  a  building 
worth  $7500  at  the  rate  of  24  ct.  for  $250  ? 

—  x  $7500  =  $7.20. 
250 


EXAMPLES   XCVI. 
Written    Exercises. 

1.  After  paying  5%  to  his  agent,  a  man  received 
$  1436.40;  what  was  the  agent's  commission? 

2.  What  is  the  amount  of  annual  premium  for  the  in- 
surance of  a  building  for  $8520  at  -fa%? 

3.  A  landlord  allowed  his  tenants  20%  reduction  from 
their  rents;  what  was  the  nominal  rent  of  a  tenant  whose 
reduced  rent  was  $1800  ? 

4.  A  commission  merchant  sells  goods  for  $2864  and 
sends  to  his  principal  $2824.62  after  deducting  com- 
mission ;  what  was  the  %  commission  ? 

5.  A  commission  merchant  is  asked  to  purchase  $6800 
worth  of  goods  at  2|%  commission;  how  much  money 
was  paid  by  his  principal  ? 

6.  A  commission  merchant  received  $6953  with  which 
to  purchase  goods  after  deducting  2\  %  commission ;  what 
was  paid  for  the  goods  ? 

7.  An  agent  sold  goods  for  $5672;  his  bill  for  ex- 
penses was  $56.72,  and  his  commission  was  11% ;  what 


f0  of  the  selling  price  did  the  principal  receive  ? 


Art.  225.]  TAXES    AND   DUTIES.  241 

8.  A.  man  insured  his  life  for  $5000  at  an  annual 
premium  of  2|-% ;  how  much  had  he  paid  at  the  end  of 
13  years  ? 

9.  A  cargo  is  insured  for  $254500,  its  full  value,  at 
2% ;  the  ship  is  insured  for  $120000  at  2|% ;  the  owner 
of  the  cargo  pays  all  insurance  and  sells  his  goods"at  the 
end  of  the  voyage  at  an  advance  of  9%  over  total  cost, 
allowing  $2000  for  freight;  what  was  the  selling  price? 

10.  The  premium  for  insuring  a  building  at  2J%  is 
$1136.25;  find  the  insurance. 

11.  A  company  insured  a  building  and  the  goods  it 
contained  for  $117944,  the  goods  being  worth  15%  of  the 
value  of  the  building.  The  merchant  paid  2%  premium 
on  the  building  and  \\%  premium  on  the  goods;  what 
was  the  total  premium  ? 

12.  A  man  sold  through  an  agent  some  merchandise, 
paying  the  agent  5%  commission.  The  agent  invested 
the  proceeds  in  two  parts  after  taking  out  commissions 
of  $325  at  5%,  and  $260  at  4%,  respectively;  what  was 
the  value  of  the  merchandise  ? 

13.  A  man  had  two  houses,  each  costing  $5000;  he 
insured  one  for  $4000  at  \\%  and  the  other  for  $6000 
at  \\°fo  :  find  the  difference  between  the  loss  on  one  and 
the  gain  on  the  other,  both  houses  having  been  burned 
on  the  day  after  insurance. 


Taxes  and  Duties. 

225.  Persons  owning  property  or  importing  goods,  pay 
to  the  government  (for  its  support)  a  certain  per  cent  of 
their  property  or  of  the  foreign  value  of  the  goods  im- 
ported. 

The  percentages  paid  on  property  are  called  Taxes. 


242  PERCENTAGES.  [Chap.  X. 

The  percentages  paid  on  imported  goods  are  called 
Duties. 

Duties  levied  on  articles  regardless  of  their  value  are 
called  Specific  Duties. 

Duties  levied  at  a  certain  per  cent  on  the  foreign  values 
of  goods  are  called  Ad  Valorem  Duties. 

In  some  States  voters  pay  annually  a  small  fixed  sum 
of  money  ($1.50  or  $2)  before  they  can  vote.  Such 
money  is  called  a  Poll  Tax 

EXAMPLES  XCVII. 
Written  Exercises. 

1.  The  expenses  of  a  certain  town  are  $39512.32 
annually ;  the  tax  is  16  mills  on  the  dollar ;  what  is  the 
value  of  the  town  as  fixed  by  the  assessors  ?  (The  asses- 
sors' valuation  is  much  smaller  than  the  real  valuation.) 

2.  The  valuation  of  a  certain  town  is  $6495860,  while 
the  assessed  valuation  is  25  %  of  that ;  the  polls  number 
1112,  and  the  taxes  are  $16.25  on  each  thousand  of  as- 
sessed valuation ;  what  are  the  expenses  of  the  town  ? 

3.  The  expenses  of  a  city  are  $339000,  and  the  assessed 
valuation  is  $16950000 ;  what  is  the  tax  rate  expressed 
as  per  cent  ?     Expressed  as  dollars  on  a  thousand  ? 

4.  What  is  the  duty  on  5000  bbl.  of  hydraulic  cement 
at  8  ct.  per  bbl.  ? 

5.  What  is  the  duty  on  125  plates  of  polished  un- 
silvered  glass  24  x  30  in.,  at  8  ct.  per  sq.  ft.  ? 

6.  What  is  the  duty  on  100  doz.  penknives  valued  at 
30  ct.  per  doz.,  at  25  %  ad  valorem  ? 

7.  What  is  the  duty  on  3 1.  of  No.  23  steel  wire  at 
2ct.  per  lb.? 


Art.  225.]  EXAMPLES.  243 

8.  A  merchant  imported  1550  yd.  of  tapestry  carpet 
valued  at  80  ct.  a  yd. ;  what  was  the  duty  at  42\  %  ad 
valorem  ? 

9.  An  invoice  of  150  doz.  linen  collars  valued  at  $  1.30 
per  doz.,  calls  for  how  much  duty  at  30  ct.  and  30  %  ? 

10.    What  does  the  government  receive  on  an  impor- 
tation of  1000  gross  of  steel  pens  at  8  ct.  per  gross  ? 


244  INTEREST.  [Chap.  XL 


CHAPTER   XI. 

INTEREST. 

Promissory  Notes. 

226.  When  one  person  borrows  money  from  another 
person,  he  gives  to  the  lender  a  written  promise  to  repay 
the  money  and  to  pay  also  a  percentage  on  the  money  at 
a  given  rate  °/0  per  year.  This  percentage  is  called 
Simple  Interest,  or  Interest. 

The  form  of  note  given  in  the  following  pages  is  the  form  in  use 
by  the  best  business  men  in  the  United  States.  Students  are 
strongly  advised  to  adhere  closely  to  the  form  while  practising  the 
making  of  notes. 

The  written  promise  is  called  a  Promissory  Note.  For 
example : 


fU-67^.  of(MJiA>iUe,,  &MH,.,  fan.  /,   18  W- 

3AaaLu  debus,  alt&v  otat&  <J  promise  to  pay  to 
the  order  of^^^~^^fcvnve&  tZH&rv^^^^^^.^^ 
^^^^onA,  IvuM/cL'i&cL  QsVxJsu-Qs&v-&yv^^„  ^  Dollar s 

Value  Received,  w-vttv  vnt&v&oZ. 
No.  82.      Due  fam,.  d//&6>.  S,  'f*. 


Arts.  226-229.]  PROMISSORY  NOTES.  245 


f250®*-.         zfjvLVnc^UU,  TMclm,.,  feu*,,  f ,  18  <?5. 

fan  cL&vyvcwicL  J^^^^promise  to  pay  to 

the  order  of^^^^^^./fcyua^&  B&l&ksA, „ 


at tA&  (^ka/fiAyyv  c/tatLcyyial  fdank,^ 

Value  received,  waXA  Cnt&^&oZ  at  5%. 

No.  763.      Dae. j.  @.  ftoumwiondL. 


227.  In  the  case  of  a  promissory  note,  it  is  to  be  noticed 
that  the  heading  indicates  the  names  of  the  Town  and 
State  in  which  the  note  is  written,  also  the  month,  day,  and 
year.  At  the  left  is  written  in  figures  the  sum  of  money 
for  which  the  note  is  given. 

228*  The  face  of  the  note  indicates  the  names  of  the 
parties  (Maker  and  Payee)  to  the  note,  the  words  'value 
received,  the  sum  of  money  for  which  the  note  is  given 
(written  in  full),  and  the  time  for  which  the  note  is  to 
run.  If  the  note  is  interest-bearing,  it  must  have  the 
words  i  with  interest '  written  in  the  face. 


).  Notes  written  as  above  may  be  transferred  (sold) 
by  the  Holder  to  another  person  (who  in  turn  becomes  the 
holder),  and  are  therefore  called  Negotiable  notes.  When 
the  note  contains  the  words  'or  order,'  the  holder  must 
Endorse  his  name  on  the  back  of  the  note  (thus  becoming 

*  The  Maker  of  a  note  is  the  person  who  signs  the  note.  The 
Payee  is  the  person  to  whom  the  note  is  made  payable.  The  Holder 
of  a  note  is  the  person  who  owns  the  note  (the  payee  or  some 
person  to  whom  the  payee  has  sold  the  note). 


246  INTEREST.  [Chap.  XI. 

responsible  for  its  payment)  when  he  sells  the  note. 
When  the  note  contains  the  words  '  or  bearer?  no  endorse- 
ment is  legally  necessary.  If  the  note  does  not  contain 
the  words  *  or  order?  or  l  or  bearer?  it  is  not  negotiable. 

230.  A  note  is  payable  at  the  business  office  of  the 
Maker  unless  otherwise  specified  in  the  note.  Nearly  all 
notes  specify  the  place  of  payment. 

231.  When  the  rate  of  interest  is  not  written  in  the 
note,  as  in  the  first  of  the  above  notes,  the  law  of  the 
State  in  which  the  note  is  to  be  paid  fixes  the  rate.  If  the 
parties  interested  wish  a  rate  different  from  the  rate  in 
the  State  in  which  the  note  is  to  be  paid,  such  rate  must 
be  specified  in  the  body  of  the  note,  as  in  the  second  of 
the  above  notes,  but  may  not  be  more  than  the  maximum 
allowed  by  such  State. 

232.  When  the  words  '  with  interest '  are  omitted  from 
a  note,  no  interest  is  payable  on  that  note  except  for  the 
time  it  may  over-run.     [Art.  239.] 

233.  The  first  of  the  above  notes  is  called  a  Time  note ; 
the  second,  a  Demand  note. 

A  time  note  is  nominally  due  at  the  date  indicated  in 
the  note,  but  Matures  (becomes  legally  due)  three  days 
later.  The  three  days  are  called  Days  of  Grace.  In  some 
States  no  days  of  grace  are  allowed. 

234.  When  a  note  matures  on  a  Sunday  or  on  a  legal 
holiday,  it  is  payable  in  some  States  on  the  business  day 
next  preceding,  and  in  other  States  on  the  business  day 
next  succeeding,  such  Sunday  or  legal  holiday. 

235.  The  dates  on  which  a  note  is  nominally  and 
legally  due  are  indicated  thus :  Feb.  8/11,  1880. 


Arts.  230-238.]  TABLE   OF  KATES.  247 

If  the  time  of  payment  is  indicated  in  '  days  after  date,1 
those  days,  together  with  three  days  of  grace  (if  such  be 
allowed  by  the  State),  are  counted  forward  from  (not 
including)  the  date  of  the  note  in  finding  the  date  of 
maturity;  thus,  the  first  of  the  above  notes  matures 
Jan-31/Feb.3,  1894. 

236.  If  the  time  of  payment  is  indicated  in  'months 
after  date]  calendar  months,  together  with  three  days  of 
grace,  are  counted  forward  from  the  date  of  the  note  in 
finding  the  date  of  maturity ;  thus,  the  maturity  of  a  note 
for  2  months,  dated  Jan.  31,  1892,  was  Mar-  31/Apr.  3,  1892 ; 
for  3  months,  it  was  APr-  30/May  3,  1892 ;  for  one  month,  it 
was  Fob.  29/Mch>8>  1892. 

237.  If  payment  of  a  note  is  not  made  on  the  day  of 
maturity,  the  holder  must  engage  a  Notary  Public  to  send 
to  the  endorser  (or  endorsers)  a  written  notice  of  such  fact. 
This  notice  is  called  a  Protest.  The  protest  must  be  sent 
on  the  day  of  maturity,  otherwise  the  endorser  cannot  be 
held  to  the  payment  of  the  note. 

Table  of  Rates  of  Interest. 

238.  The  following  table  gives,  for  each  of  the  States 
and  Territories,  the  Legal  Bate  when  no  rate  is  mentioned 
in  a  note,  the  Maximum  Kate  allowed,  the  Time  of  pay- 
ment when  the  day  of  maturity  falls  on  a  holiday  (the 
day  before  by  B,  and  the  day  after  by  A),  and  indicates 
by  the  letter  G  those  States  in  which  days  of  grace  are 
legal. 

Notes  made  on  or  after  Jan.  1,  '95,  and  payable  in  New 
York,  bear  no  grace. 

Notes  made  after  July  4,  '95,  and  payable  in  New  Jer- 
sey, bear  no  grace. 


248 


INTEREST. 


[Chap.  XI. 


State. 

"S 

i 

• 

a 

e 
6 
1 

State. 

c3 

M 

03 

2 

M 

% 

H 

8 

M 

a 

H 

o 

Alabama    .     .     . 

8 

8 

A. 

<;. 

Montana  .    .     . 

10 

Any. 

B. 

G. 

Arizona     .    .     . 

7 

Any. 

A. 

G. 

Nebraska .    .    . 

7 

10 

A. 

G. 

Arkansas  .     .     . 

6 

10 

B. 

G. 

Nevada     .    .     . 

7 

Any. 

B. 

G. 

California  .     .     . 

7 

Any. 

A. 

New  Hampshire 

6 

6 

B. 

G. 

Colorado    .    .     . 

8 

Any. 

B. 

G. 

New  Jersey .     . 

6 

6 

A. 

Connecticut  .     . 

6 

6 

B. 

G. 

New  Mexico 

6 

12 

A. 

G. 

Delaware  .     .     . 

6 

6 

B. 

G. 

New  York    .     . 

6 

6 

A. 

Dist.  of  Columbia 

6 

10 

A. 

G. 

No.  Carolina 

6 

8 

B. 

G. 

Florida  .... 

8 

10 

B. 

G. 

No.  Dakota  .    . 

7 

12 

A. 

G. 

Georgia     .    .     . 

7 

8 

A. 

G. 

Ohio     .    .    .    . 

6 

8 

B. 

G. 

Idaho    .... 

10 

18 

A. 

Oklahoma     ,     . 

7 

12 

B. 

G. 

Illinois.     .     .     . 

5 

7 

B. 

G. 

Oregon     .    .     . 

8 

10 

A. 

Indiana  .     .     .     . 

6 

8 

B. 

G. 

Pennsylvania    . 

6 

6 

A. 

G. 

Indian  Territory- 

6 

10 

B. 

G. 

Rhode  Island    . 

6 

Any. 

A. 

G. 

Iowa     .... 

6 

8 

B. 

G. 

So.  Carolina  .     . 

7 

8 

A. 

G. 

Kansas .... 

6 

10 

B. 

G. 

So.  Dakota. 

7 

12 

A. 

G. 

Kentucky .     .     . 

6 

6 

B. 

G. 

Tennessee 

6 

6 

B. 

G. 

Louisiana  .     .     . 

5 

8 

A. 

G. 

Texas  .    . 

6 

10 

B. 

G. 

Maine    .... 

6 

Any. 

B.orA. 

G. 

Utah    .     . 

8 

Any. 

B.orA. 

Maryland  .     .     . 

6 

6 

B. 

G. 

Vermont  . 

6 

6 

A. 

Massachusetts    . 

6 

Any. 

A. 

G. 

Virginia    . 

6 

6 

B. 

G. 

Michigan   .     .     . 

6 

8 

A. 

G. 

Washington 

8 

Any. 

B. 

G. 

Minnesota      .    . 

7 

10 

A. 

G. 

W.  Virginia 

6 

6 

B. 

G. 

Mississippi     .    . 

6 

10 

B. 

G. 

Wisconsin 

6 

10 

A. 

Missouri    .     .     . 

6 

8 

A. 

G. 

Wyoming 

12 

Any. 

A. 

G. 

EXAMPLES   XCVIII. 

1.  Write  a  time  note  for  $  250.67  with  interest  at  16  %. 

2.  Write  a  time  note  for  $  76  with  interest  at  20  %. 

3.  Write  a  time  note  for  $468.92  for  20  da.  without 
interest. 

4.  Write  a  time  note  for  $20  for  4  mo.  with  interest 
at  13%. 

5.  Write   a  time  note  for   $560,  headed   Cincinnati, 
Ohio,  Jan.  13th,  1892,  to  mature  in  63  da.,  with  interest. 

6.  Write  a  demand  note  for  $528  with  interest. 

7.  Write  a  demand  note  for  $460  with  the  maximum 
interest  allowed  by  the  State  in  which  you  live. 


Arts. 

239,  240.]          SIMPLE   INTEREST. 

249 

Find  the  date  of  maturity  of  each  of  the  following 

indies 

ited  notes : 

Date. 

Where  Payable. 

Time. 

8. 

Dec.    18,1895 

New  York 

30  days 

9. 

Jan.    18,  1895 

New  York 

60  days 

10. 

Jan.    27,1896 

New  York 

45  days 

11. 

July  31,  1895 

New  Jersey 

60  days 

12. 

June    6,  1895 

New  York 

3  months 

13. 

Jan.    30,1895 

New  York 

1  month 

14. 

Jan.   30,  1896 

New  York 

1  month 

15. 

Mar.  31,  1895 

New  York 

3  months 

16. 

Mar.  31,  1895 

Conn. 

1  month 

17. 

Mar.  30,  1896 

Mass. 

1  month 

18. 

Sept.    3,  1890 

Nebraska 

60  days 

19. 

Jan.   29,  1896 

Cal. 

30  days 

20. 

Jan.   29,  1896 

Cal. 

1  month 

Simple  Interest. 

239.  Interest  on  the  Principal  (money  borrowed)  is 
called  Simple  Interest,  and  is  computed  at  the  given  rate 
per  cent  (per  year  understood)  for  the  time  elapsing 
between  the  date  and  maturity  of  the  note. 

If  a  note  is  not  interest-bearing  and  is  not  paid  at  maturity, 
interest  is  payable  after  maturity  and  until  the  note  is  paid. 
[Art.  232.] 

240.  The  majority  of  notes  are  given  for  short  periods 
of  time  —  say  30,  60,  or  90  days,  or  1,  2,  or  3  months. 
Now  it  is  customary  in  interest  computations,  to  regard 
one  year  as  360  days.  Therefore,  by  a  short  operation, 
we  may  find  the  interest  on  any  principal  for  any  time 
and  at  any  rate  per  cent. 


250 


INTEREST. 
Ex.  1. 


[Chap.  XL 


$350^-.  tfJUxwuif,  o/t.y.,  fan,-  /6,  18<?6. 

€Lakt&eA/L  cLa^uoy  alt&v  cCat&  of  promise  to  pay  to 
the   order  of jo-el  cPuC-, 


,ncwyi>^ 


.qI^iv&&  hnA/yvcLi&cL 


.^Dollars 


Value  received,  w-ttk  ImZ&v&oZ. 

JVo.  68.      Due  &e&.  S.  d.  &.  g&nMe, 


350 

.06 


360)  $21.00 
.051 

18 


$1.05 


In  this  note,  the  rate  is  6%  and  the  time  is  18 
da. ;  the  interest  for  1  yr.  will  be  .06  of  the  princi- 
pal, —  $21 ;  the  interest  for  one  day  is  found  by 
dividing  $21  by  360,  and  the  interest  for  18  days 
is  found  by  multiplying  the  quotient  thus  found 
by  18.  Hence,  .06  and  18  are  multipliers,  while 
360  is  a  divisor. 


This  may  be  expressed  as  follows 
350  x  6  x  18 


100  x  360 
.350  x  18 


which  becomes 


by  cancellation. 


We  observe  that  interest  for  any  number  of  days  may  be  found 
by  dividing  the  principal  by  1000,  multiplying  by  the  number  of 
days,  and  dividing  by  6. 

The  following  is  a  better  form  for  practical  work : 


350. 


$1,050 


o  Here,  cancelling  6  from  the  dividend  and  divisor, 

we  have  .350  to  be  multiplied  by  3. 


Art.  241.]  SAMPLE   INTEREST.  251 

Ex.  1.  In  the  above  note  let  the  principal  be  $367.91,  the  rate 
6%,  and  the  time  21  da. ;  find  the  interest. 

9fffl.ffX     x18395  +  We  must  cancel  the  divisor  complete- 
rs       1  ly ;   only  5  figures  will  be  needed  for 
$1.28765  the  multiplicand  ;  keep  the  multiplier  as 
$1.29  =  Ans.  small  as  possible  by  cancellation. 

Ex.2. 


$650™-.  dufuda,,  7H&.,  fan.  7,  18<?3. 

3w4-  wumMa/  att&h  dat&  c/  promise  to  pay  to 

the  order  of^^^^^./lf&nvy>  Butted, 

^_^__c/t^  kwruLv&cl  (t^tif jjg  Dollars 

at^^^~jLh&  &iaMAt&  cAatvcyyval  fdanfo 

Value  received,  with,  (mt&v&oZ. 

/if&wvif  JS.  Reld. 
No.  152      Due  TnavsA  7//0,  '$3. 


Find  the  interest. 

01$  JJj30.         325  When  the  time  is  '  months  after 

?1       @* 21  date,'  calendar  months  are  counted 

005 
fi  ZZr  in  obtaining  the  date   of  maturity 

1 6  825  [Art.  236],  and  the  interest  is  com- 

6.83  =  interest,      puted    for   the    stated    number   of 

months  counting  30  da.  as  one  month. 

This  note  payable  in  Maine  has  three  days  of  grace  ;  therefore  the 

interest  is  computed  for  63  da. 

241.  The  value  of  a  note  at  its  date  of  maturity  is 
called  its  Maturity  Value,  and  consists  of  the  sum  for 
which,  the  note  is  given  plus  the  interest  (if  any).  In 
finding  maturity  value,  observe  whether  or  not  the  note 
bears  int.  and  '  days  of  grace.9 


252 


INTEREST. 


[Chap.  XI. 


EXAMPLES   XCIX. 


Written  Examples. 


What  interest  is  due  at 

maturity  on 

each  of  the  follow- 

g  indicated  notes  ? 

Date. 

Principal. 

Kate. 

Time. 

1.  Arkansas, 

$763 

6% 

12  da. 

2.  New  Jersey, 

$1467 

15  da. 

3.  Ohio, 

$1626.75 

18  da. 

4.  Texas, 

$6000 

21  da. 

5.  New  York, 

$5267.50 

27  da. 

6.  New  York, 

$2675 

27  da. 

7.  California, 

$376 

2  mo. 

8.  Kentucky, 

$498 

1  mo. 

9.  Connecticut, 

$75000 

108  da. 

10.  New  Hampshire, 

$704.25 

201  da. 

11.  Illinois, 

$84.75 

361  da. 

12.  Utah, 

$846 

51  da. 

242.  For  rates  other  than  6%,  find  the  interest  at  6% 
and  take  such  a  part  of  that  interest  as  the  given  rate  is 
of  6%. 

Ex.  1.  P  =  $  4673,  B  =  5%,  time  =  33  da. ;  find  interest. 


M 


2  3365 
11 


6)25.7015 
4.2835 


$21.42     =Ans. 


Here  the  interest  at  6%  is  $25.7015, 
and  |  of  this  is  $21.42.  We  obtain  this 
result  rapidly  by  subtracting  the  interest 
at  1  %  from  the  interest  at  6  %. 


Art.  242.]  SIMPLE   INTEREST.  253 

Ex.  2.   P  =  $  26.48,  B  =  1\  %,  time  =  90  da. ;  find  int. 

„  1 026.48 

ff0    15  Here  the  interest  at  7^  %  =  f  of  the  interest  at 

4)  .39720      6%.     The  answer  is  obtained  by  adding  to  the 

•0993       interest  at  6  %  one-fourth  of  the  interest  at  6  %.* 
$.50 

In  the  j'inaZ  work  do  not  waste  time  writing  anything  but  the 
answer. 

EXAMPLES  C. 


)S 


fYOO0-^.  RaUicjk,  <A.&.,  0Lucf.3r,  18  <?5. 

&hi&&  rnxyyitko/  alt&v  dat&  <J  promise  to  pay  to 
the  order  of^^^^^^^^ko-Y/ia^  $v&&n>&. 

s^^^^^.$/&v-&n  kuncCt&cC 

at tk&  @Ctiq&H&>  c/tatvcyyial  Bank, 

Value  received,  w\tk  vnt&v&oA,  at  7  °lo. 

No.  16      Due  cAov-.  SO / '£&&    3,  /8<?5. 


g  Dollars 


1.  Find  the  interest  on  the  above  note. 

2.  Find  what  would  be  the  amount  of  the  above  note 
if  the  time  were  90  da. 

3.  Find  interest  on  a  New  York  note  for  $  2670,  dated 
Dec.  31,  1895,  payable  47  da.  after  date,  with  interest. 


*  Eor  interest  at  8  %  add    }  of  interest  at  6  %    to    itself. 

710/  U  1 

'2/0  f 


The 


70/  11  1 
5|-%subt.TV 
5  %  "  i 
H%  "  i 
4  %  "  I 
3   %    =      i 


from 


254 

INTEREST. 

[Chap.  XI. 

What  interest  is  due  at 
lowing  indicated  notes  ? 

;  maturity 

on  each  of  the  fol- 

4. 

Date. 

Saratoga,  N.Y., 

Principal. 

$2670 

Eate.         Time  after 
Date. 

Maximum.  47  da. 

5. 

Springfield,  Mass., 

$4893 

5% 

60  da. 

6. 

Washington,  D.C., 

$289.20 

6% 

90  da. 

7. 

Buffalo,  KY., 

$48.93 

H% 

3  mo. 

8. 

Utica,  N.Y., 

$48.93 

4% 

3  mo. 

9. 

Baltimore,  Md., 

$765 

Maximum.    6  mo. 

10. 

Hartford,  Conn., 

$4893 

it 

90  da. 

11. 

Denver,  Col., 

$8695 

12% 

24  da. 

12. 

St.  Louis,  Mo., 

$463.50 

9% 

60  da. 

13. 

Chicago,  111., 

$873 

61% 

30  da. 

14. 

St.  Paul,  Minn., 

$  487.20 

m 

2  mo. 

15. 

NewYork,N.Y., 

$286.37 

4f% 

3  mo. 

16. 

Boston,  Mass., 

$  499.99 

4% 

60  da. 

17. 

Find    the    amount 

in    each 

of    the 

last   three 

examples. 

The  above  method  for  computing  interest  is  in  general 
use  when  the  time  is  less  than  one  year ;  but  if  the  time 
is  in  yr.,  mo.,  and  da.,  the  6%  method  is  the  more 
frequently  used. 

Six  Per  Cent  Method. 

243.  A  demand  note  for  $  268.50,  dated  Nov.  26,  >87, 
was  paid  4  yr.  7  mo.  18  da.  after  date ;  what  was  the 
interest  at  6%  ? 


Arts.  243, 244.]  SIX  %  METHOD.  255 

At  6% 

the  interest  on  $1  =  $.06  (six  cents)  for  1  yr., 
«  "  $1  =  $.005  (5  mills)  for  1  mo., 

"         "  $1  =  $.0001  (I  of  a  mill)  for  1  da. ; 


u 

hence 
u 


«  «  $1  =  $.24  for  4  yr., 

"  «  $1  =  $.035  for  7  mo., 

"        "  «  $1  =  $.003  for  18  da., 

"         «  "  $1  =  $.278  for  4  yr.  7  mo.  18  da. 

$  268.50         Having  found  the  interest  on  $  1  for  the  given  rate 
.278    and  time,  we  multiply  this  interest  by  the  principal. 
$   74.64     [Art.  47,  Theorem  I.] 

Note.  It  is  evident  that  the  interest  for  2  mo.  at  6  %  may  be 
computed  by  moving  the  decimal  point  2  places  to  the  left.  Thus, 
the  interest  on  $784.70  for  2  mo.  is  $7.85.  Similarly,  the  interest 
for  6  da.  is  $.78.  Also  the  interest  for  12  da.  is  $1.57.  The  6  % 
method  is  sometimes  used  when  the  times  are  less  than  1  yr. 

244.  There  is  great  diversity  in  the  methods  of  finding 
the  time  in  a  case  like  this.  Some  prominent  banks  and 
business  houses  in  the  United  States  use  the  method  of 
counting  the  time  in  years  and  days,  instead  of  the  method 
just  described. 

Thus,  the  above  note  was  paid  July  14,  '92;  the  4yr.  were 
counted  forward  from  Nov.  26,  '87,  the  7  mo.  were  counted  for- 
ward as  calendar  months  from  Nov.  26,  '91,  and  the  18  da.  were 
counted  forward  from  June  26,  '92.  (This  is  not  compound 
addition.) 

In  obtaining  the  int.  the  years  were  reckoned  as  wholes,  but  the 
months  and  days  were  reckoned  in  the  exact  number  of  days 
found  in  that  7  mo.  and  18  da.  which  began  with  Nov.  26,  '91.  Thus, 
the  time  was  4  yr.  231  da.  —  4  in  Nov.,  31  in  Dec,  31  in  Jan.,  29  in 
Feb.,  31  in  Mar.,  30  in  Apr.,  31  in  May,  30  in  June,  14  in  July. 
The  int.  on  $1  =  $.24  for  4  yr., 
"  "  $1  =  $.0385  for  231  da., 

"  $1  =  $.2785  for  4  yr.  231  da. 
$268.50  x  .2785  =  $74.77  =  Ans. 


256  INTEREST.  [Chap.  XI. 

EXAMPLES   01. 
Written  Exercises. 

What  was   the   amount   at  maturity  of  each   of  the 
following  indicated  notes  ? 

Obtain  answers  by  each  method  [Arts.  243,  244]. 


Date. 

Principal. 

Rate 

Time. 

1. 

Philadelphia 

July  31, 

'84 

$7680.95 

6% 

3yr. 

6  mo.  12  da. 

2. 

Richmond, 

Aug.  6, 

'87 

$683.42 

4yr. 

7  mo. 

3. 

Boston, 

Jan.  9, 

'88 

$1492.88 

4yr. 

11  mo.  18  da. 

4. 

Cleveland, 

Oct.   1, 

'90 

$2689.42 

2yr. 

10  mo.  18  da. 

5. 

Jersey  City, 

Aug.  1, 

'95 

$487.50 

lyr. 

7  mo.  13  da. 

6. 

Providence, 

Aug.  8, 

'79 

$2000 

8yr. 

5  mo. 

245,   For  rates  other  than  6%,  proceed  as  in  Art.  242. 

EXAMPLES   OIL 

A  few  demand  notes  are  here  indicated;  find  the 
interest  on  each  of  the  first  four,  and  the  amount  on  each 
of  the  others. 

Obtain  answers  by  each  method. 


Date. 

Principal. 

Rate. 

Paid  after  Date. 

1. 

June  8, 

'81 

$468.93 

41  o/ 
*J  /o 

5  yr.  9  mo.  18  da. 

2. 

Aug.  2, 

'87 

$1680.50 

4    » 

3yr.  6  mo.  27  da. 

3. 

Feb.  29, 

'88 

$2500 

5i" 

3yr.  11  mo.  6  da. 

4. 

Mch.  9, 

'91 

$155 

B|" 

lyr.  3  mo.  3  da. 

5. 

May  13, 

89 

$450.50 

*h" 

3  yr.  2  mo.  7  da. 

6. 

Oct.  23, 

'85 

$896.88 

4£" 

4yr.  11  mo.  29  da 

7. 

Sept.  15 

'82 

$15875 

2|" 

9yr.  6  mo.  15  da. 

Annual  Interest. 

246.  Some  States  allow  interest  to  be  collected  on  each 
annual  instalment  of  interest,  if  such  instalment  is  not 
paid  when  due. 


Art.  245, 246.]  ANNUAL   INTEREST.  257 

Ex. 


$673^.  Bwoktyn,,  ofi.y.,  CtfriM  6,  18  <?5. 

€n  oLeAYvancL  J ^^^^^^jpromise  to  pay  to 

the  order  of ftewuy  ofyyvltk 

^^^cfwc  AsWvuOi&ct  Qs&v-eMXAf-tkv&z-^-^^DoTlcLrs 
Value  received,  with  iAtt&i&oZ  a/yuriMaZVu  at  3  % . 
No.  /7>      Due Jd&nja-vyvLrv  jcyuLan. 


If  this  note  be  paid  in  3  yr.  6  mo.  after  date,  and  no  interest  has 
"been  paid  meanwhile,  there  will  be  paid  the  principal,  the  simple 
interest  on  the  principal,  and  simple  interest  on  each  annual  instal- 
ment of  interest  from  the  time  it  is  due  until  the  note  is  paid. 


.21 


6)141.33 
23.555 

$117,775  =  interest  for  3  yr.  6  mo. 
7.571  =  interest  on  interest. 
673.        =  principal. 
$798.35    =  amount  to  be  paid. 

The  1st  instalment  bears  interest  for  2  yr.  6  mo. 
The  2d  instalment  bears  interest  for  1  yr.  6  mo. 
The  3d  instalment  bears  interest  for  6  mo. 

Interest  on  annual  instalment 

is  computed  for  4  yr.  6  mo. 

$33.65  =  annual  instalment  at  5  %. 
.27 

6)9.0855 
1.5142 


$7,571    =  interest  on  interest  for  4  yr.  6  mo. 


258  INTEREST.  [Chap.  XL 


EXAMPLES  CIII. 

Written  Exercises. 

1. 


$/86d^..  Jt&w-  Tfovk,  o/t.lf,  fan,  /a,  18  <?&. 

€.n  oLeAn&nci  J-^^^^jpromise  to  pay  to 

€vcfkt&&vi  hwyicOvect  &v?ctu-thi,&&  cwici  ^Dollars 
Value  received,  w\Zk  Cnt&i>&aZ  a/yt/yt/waMu  at^-%. 
No.  V-/7.      Due. /if&nvif 


If  this  note  be  paid  5  yr.  8  mo.  and  20  da.  after  date, 
no  interest  being  paid  meanwhile,  how  much  will  the 
holder  receive  ? 

2.  Cast  the  interest  on  a  note  similar  to  the  above, 
when  P=  $897.75,  R  =  h\%,  and  r=4yr.  9 mo.  and 
15  da.,  no  interest  being  paid  meanwhile. 

3.  How  much  does  a  man  owe  at  the  maturity  of  a 
note  similar  to  the  above,  when  P=  $437.25,  i2  =  4%, 
and  T  =  7  yr.  27  da.,  no  interest  having  been  paid  ? 

Commercial  Discount. 

247.  We  have  been  considering,  in  the  last  few  pages, 
cases  in  which  money  is  borrowed  from  persons ;  we  have 
learned  that  the  interest  is  payable  at  the  maturity  of  the 
note. 


Arts.  247-249.]     COMMERCIAL  DISCOUNT.  259 

When  money  is  borrowed  from  a  bank,  the  interest 
(simple)  is  paid  on  the  day  on  which  the  money  is  borrowed. 

The  simple  interest  which  a  bank  takes  in  advance  is  y 
called  Commercial  Discount,  or  Bank  Discount. 

The  borrower  does  not  receive  the  principal  (as  when 
borrowing  from  a  person),  but  receives  the  principal  minus 
the  simple  interest  on  the  principal;  this  remainder  is 
called  the  Proceeds  of  the  note. 

The  following  example  will  show  the  methods  of  calcu- 
lation of  discounts  and  proceeds. 

Ex.  1. 


n,  TWouM,.,  c/tav-.  16,  18  qy-. 

^kwiUf  clwyo,  ajt&v  dat&  <J  promise  to  pay  to 
the  order  of wjm*IL~~~ 

<^W  Aundv&cl  Q,v?cty-Q,&v-&n — ~~  ^  Dollars 

at^^^^^£h&  S^OlqZ  <Aa,tu>naZ  JSc^rik^^^^^^ 
Value  received. 

No.  /?.     Due  be*.  /6//f,  '<?£. 


248.  When  a  person  borrows  money  from  a  bank,  he  makes  his 
note  payable  to  himself,  and  at  the  bank  which  makes  the  loan. 
The  note  must  be  endorsed. 

249.  The  discount  is  computed  on  the  maturity  value 
of  the  note ;  in  the  above  note  it  is  computed  on  $267, 
since  the  note  is  not  interest-bearing.  So,  also,  when  a 
person  sells  a  note  to  a  bank  (Ex.  2,  following),  the  bank 
discounts  its  maturity  value. 

Find  the  discount  and  proceeds  of  the  note  in  Art.  247. 


260  INTEKEST.  [Chap.  XI. 

0  |  J20J.     .  1335  It  will  be  observed  that  in  case  money 

H  is  borrowed  from  a  person  the  borrower 


$1.4685  has  the  use  of  a  larger  sum  of  money 

$267  00  than  when  he  borrows  from  a  bank,  yet 

1.47  =  discount.         ne  PaYs  tne  same  interest.     The  bank 


$265.53  =  proceeds.         nas  tlie  use  of  tne  discount  while  the 

borrower  has  the  use  of  the  proceeds. 
At  the  maturity  of  a  note  given  to  a  person  the  borrower  pays 
principal  and  interest ;  at  the  maturity  of  a  note  given  to  a  bank 
the  borrower  pays  the  principal  only,  having  already  paid  the 
interest. 

Ex.  2. 


#700- .  c/tew  tfcuv-sM,,  @syvwi.,  fcvn.  3/,  18  <?5. 

fai&  wuynXA  a^C&v  ciat&  J  promise  to  pay  to 

the  order  of fowu&o,  (^kiAUf~~~~~~ 

^v^vvvvvvv^ofeiwt  Au^c^i^d^^^^^^^YQQ  Dollars 
at  tA&  ofv^t/i  (Zv-&nu&  Bam^k,  cA&w  llo-vk,  o/Jf.ll. 
Value  received,  with,  vyit&v&oZ  at  ¥i%. 
No.  27.      Due  Ss6.  28/ q  5.  ftenvif  3iU&. 


Discounted  Feb.  5,  at  6%. 

In  this  case  the  holder  took  a  promissory  note  to  some  bank, 
and  the  bank  discounted  the  note  ;  i.e.,  the  teller  gave  the  holder 
the  proceeds  of  the  note  calculated  on  its  maturity  value.  The 
time  for  which  a  bank  computes  discount  is  the  exact  number  of 
days  from  the  day  of  discount  to  the  day  of  maturity,  although 
the  time  of  the  note  may  be  written  in  months.  The  holder  must 
endorse  the  note.  [Art.  229.] 
flflJOO. 


4)3.50 
.875 
$2.63  =  interest  at  4i%. 
$702.63  s  maturity  value. 


1         23 

.1171 
23 

$2.69  =  discount. 

$702.63 
2.69 

$699.94 

=  proceeds. 

Art.  249.]  COMMERCIAL  DISCOUNT.  261 

The  note  matures  without  grace,  because  payable  in  N.Y.    The 
term  of  discount  is  the  number  of  days  from  Feb.  5  to  Feb.  28. 

Ex.  3. 


f<?78  —  ,  BuAl\#ujL<yn,,  Vt.,  (let.  /2,  18  <?#■. 

&ia>-&  wvonZAa,  a^t&v  clat&  I  promise  to  pay  to 

the  order  of 3%&c£&vi<&fc  /i-o&wv&'b 

^^jcAvyi&  kwncOi&ci  &&v-&nty-&tf  A  t-^^YQo  Dollars 
at^J^^^JJv&  H-ow-oaxL  cftatLo-yicil  Bavifc^^^s^ 
Value  received,  w-iXA  int&v&at. 
No.  /  7.     Due  TftaA&k  12,  '  <?5.      ft&wuu  Shovta*. 


Discounted  Dec.  31,  '94,  at  5%. 

01978.25  a\  1002.71         .16712 

I         m    25  P|  71  71 

$24,456  6)11.86552 

$978.25  1.97758 

$1002. 71  =  maturity  value.  $9.89  =  discount. 

$1002.71  -  $9.89  =  $992.82  =  proceeds. 

Ex.  4.   A  merchant  wishes  $750  for  immediate  use  for  60  days. 
What  must  be  the  principal  of  his  note  given  to  a  New  York  bank  9 

Here  we  have  the  proceeds  and  rate  given,  to  find  the  principal. 
Find  the  proceeds  of  $  1  for  60  days  ;  this  will  be  $  .99.     Then, 
proceeds  of  $1 :  given  proceeds  :  :  $1  :  principal. 

.-.  Required  principal  =  given  Proceeds 
proceeds  of  $1 

=  11^0  =  $757.58. 
$.99 


262  INTEREST.  [Chap.  XL 

EXAMPLES   CIV. 
1. 


s 

f8700™-.          o/tcuAvUU,,  &wn,.,  &e&.  27, 18  ft. 

^ 

cALnetif  cLaAf&  a/^t&v  cLat&  J  promise  to  pay  to 

<jj$ 

the  order  of^^^^^^^^^m^etl 

8 

€Lykt  tkow&cwicL  &&v-&ri  hawicOb&cL ^Dollars 

d| 

at tk&  BunJc  a^  (7LcmvmsEA&& 

m 

Value  received,  ivaXsJv  imt&vejoZ. 

3 

No.  /J?  7.      Due flfemAAf  £fvm&. 

Find  maturity,  discount,  and  proceeds. 

The  State  in  which  a  note  is  to  be  paid  determines  the 
question  of  '  Days  of  Grace.' 


$3000^-.  Jb&MAWU,  &ol.,  fan.  6,  18  <?2. 

^vxjju  cLoAfo,  a^t&i  dat&  J  promise  to  pay  to 

the  order  of^^^r~~*~^wiAftoe>lfy 

c^iifi&  tAowxMui^^^^^^.^  Dollars 

at tk&  c/tavtk  £&nv-&v  BamJo. 

Value  received. 

No.  70.      Due ft&vnux/H, 


Find  maturity,  discount,  and  proceeds, 


Art.  249.] 


EXAMPLES. 


263 


fV-530^-. 


3. 


Sfcvwta,  Si,  Jt.  m. ,  &e&.  28,  1 8  <?3. 


3kv&&  rnxmtko,  oL^t&v  clat&  J  -promise  to  pay  to 

the  order  of^^^^^^^^wAf&&tjs>~~~~^ 

c^W  tAow&a'yieL  ^tv-5  kwruiv&cC  tkuity  jjjjj  Dollars 

Value  received. 

JVo.  86.      Due d/moo,  Zfu&fceA,. 


Find  maturity,  discount,  and  proceeds. 

In  all  notes  Date  of  Maturity  and  Rate  are  the  first 
things  to  consider. 


3w&  WLO-ntha,  a^t&v  clat&  J  promise  to  pay  to 
the  order  of^^~^~~—^> 

at^^^^^£k&  cAatian^l  Ji/yrv  fdasvifc. 

Value  received. 

No.  /<?-      Due Ww,.  fC.  7\W^mW. 


^Dollars 


Find  maturity,  discount,  and  proceeds. 


264  INTEREST.  [Chai-.  XI. 


5. 


$685™-.  JW  3>aw,  TnUk.,  CUuf.  3/,  18  <?¥. 

&tv-&nXAf  elaAfb  oulteAj  ciaZ&  <J  promise  to  pay  to 

the  order  of^^^~~~~~^./mjf&M 

^v^v^-.c/u^  kwyioUs&cl  &vyA£M-^vv-& ^Dollars 

aty^^^^ysy^^XA&  gPCuiZ  c/tatvcyyiaZ  fSasvik-^^^^^. 

Value  received. 

No.  //.      Due....  &.  &.  $a,vk^. 


Find  maturity,  discount,  and  proceeds. 


6. 


/  aft.  CHAom*,  Vt.,  fvutif  Sf,  18  87. 

^ev&n  mcmtho.-  cvft&v  date,  J  promise  to  pay  to 
the  order  of^ 


m  Dollars 


at^^^^Xh&  W^&ldeAv  cAatio-nai  Ba/wk~~~~~~ 

Value  received. 

No.  22.      Due...  £exyvia<u£  j?&wm&. 


Discounted  at  6%. 

Proceeds  =  $6269.25 ;  find  maturity  and  discount. 


Art.  249.]  EXAMPLES.  265 


7. 


/  cA&w*  Ifcyvk.,  cA.lf.,  fan.  /,   18  <?6. 

£fw~o-  mo-ntko,  a^t&v  claZ&  <J  promise  to  pay  to 

the  order  of^ wvif&el^ 

_ m  Dollars 

at^^~^~~Xk&  &<ym  €vc&kaM,(f&  JSank 

Value  Received. 

No.  /.     Due @Aa&.  €.  fdvown,. 


Discounted  at  4%. 

Proceeds  =  $248600 ;  find  maturity  and  discount. 

8.   What  would  have  been  the   discount  in  7,  if  the 
note  had  been  discounted  by  a  bank  in  Ohio  at  the  same 


/  Boston-,  maw-.,  CtfU.  6,  18  W. 

otvfte&n  day*  aft&v  dat&  J  promise  to  pay  to 

the  order  of mA^&tj ~~~~ 

m  Dollars 

atr^^^^£>A,&  3wloL&uq/  cAatvonal  Bcun^^^j^^^^. 
Value  received. 

No.  /8.     Due JCe&na/uL  Tyioundtt. 


Discounted  at  4^  <J0. 

Proceeds  =  $100000 ;  find  maturity  and  principal. 


266  INTEREST.  [Chap.  XL 

10. 


$3560™.        cAew  Ifovk,  Jt.y.,  tfefiL.  /<?,  18  <?5. 

<&iv-&  mo-ntA^  ait&v  ctat&  cj  promise  to  pay  to 

the  order  of &ied&vU&  <$>vvne>& 


£'/vv&&  tAowMMicL  jZv&  hwvicLv&cl  vLocLif  ^  Dollars 

at £h&  ^A&wC^clI  cAaLl(yvia,L  Ba/nfc 

Value  received,  with,  vjit&v&oZ- 

No.  56.      Due__„.  d.  f.  0LLv(yic£. 


Discounted  Dec.  31,  at  4J  %. 
Find  maturity  and  discount. 

11.  A  note  dated  N.Y.,  July  7,  1891,  payable  in  Ohio 
in  3  yr.  after  date,  was  discounted  Jan.  16, 1892,  at  4£  %  ; 
the  principal  was  $5000.     Proceeds  =? 

12. 


f780!ZL.  RUhwumd,  Va»,  fum&7,  18<?2. 

ofoutu  da,y&  a^t&v  dat&  J  promise  to  pay  to 

the  order  of^^^^^^/ifcyiatio  £ate& 

^j^^^efgAkML  fiwndv&cC  eZcfAfy  ^^^^  Dollars 

at. Xk&  ffla/nteM/ '  c^atianal  BamJo^ 

Value  received,  wiZA  Lnt&i&oZ. 

No.  27.      Due jamv&O'  ffa/nutton* 


Discounted  June  13,  at  4J%, 
Find  maturity  and  proceeds. 


Art.  249.]  EXAMPLES.  267 

13. 


f67/^-.  Jt&w-  yovk,  c/t.  If.,  fan.  20,  18  <?5. 

cA'vn&Vu  daifo,  cl^I&v  dat&  w&  promise  to  pay  to 

the  order  o/U^^ ^<xyyww&t  &volv-&& 

^^^^.c/Cpc  huM<cUi&cL  &&v-&ntu-cyyi& —Dollars 

at  the*  &Oia£  cA'atio-nal  ISc^nk,  oft.  (ZuyuoX>uyi&,  &la,. 

Value  received,  with,  Imt&v&oZ. 

No./2.     Due ftvusA,  V>  C*. 


Discounted  Feb.  28,  at  6  %. 
Find  maturity  and  proceeds. 

14. 


/  cJ'aoAua,,  c/t./f.,  £&&.  3/,  18<7¥. 

t^W  wio-ntha.'  aft&v  dat&  w-&  promise  to  pay  to 

the  order  of f3&rv[a/vnAAv  &vaA>&&^^^^. 

.___ m  Dollars 

at^^^^~^XA&  cfe^u^biXy-  3\mq£>  &>& 

Value  received,  iv~iXA  Imt&v&aZ. 

No.  /02.      Due Jbanyid  lA>cdl&. 


Discounted  Feb.  3,  at  5  %. 

Find  maturity,  maturity  value,  and  principal,  when  the 
proceeds  =  $790. 


268  INTEKEST.  [Chap.  XI. 

15. 


/ ?Hilw-oiiAJo&&,  Wio,.,  TnoAf  25,  18  <?3. 

&ewb  wo-nt/ifr  ujt&'u  dat&  J  promise  to  pay  to 

the  order  of^^^^@Aa>o,.  <P.  fdiosko-fi, 

-m  Dollars 

at^^^ — tk&  'i/'&wyLcvyL  CLryveAA^a^n  fdaswh 

Value  received,  w~\£h,  vyvteA.&oZ. 

No.  f/S.      Due /lfo-va,&&  W4vit&. 


Discounted  July  1,  '93,  at  5^%. 
Proceeds  =  $8000 ;  principals? 

Ans  j  $7920.84  =  face. 

'  1  $8110.28  =  maturity  value. 

Exact  Interest. 

250.  Thus  far  interest  has  been  computed  on  the  basis 
of  a  year  of  360  da.  Such  interest  is  evidently  \\  of  the 
interest  computed  on  the  basis  of  a  year  of  365  days. 

The  interest  computed  on  the  basis  of  a  year  of  365  da. 
is  called  Exact  Interest,  and  is  computed  for  only  fractions 
of  a  year. 

Exact  interest  is  computed  in  interest  transactions  with 
General  Governments  and  in  many  interest  transactions 
of  ordinary  business. 

Ex.  1.  Find  exact  interest  at  4i%  on  a  note  for  $892,  dated  Feb. 
16,  '93,  and  maturing  Apr.  2/5,  '93. 

Here  we  find  the  interest  at  4-*-%  for  the  exact 


4^)7  136 

i  7«d    numDer  of  days  and  on  the  360  days  basis,  and  sub 


1.784 

73)5.352 

073    answer  on  the  365  days  basis. 

$5.28 


Arts.  250,  251.]        PAKTIAL  PAYMENTS. 


269 


Ex.  2.  Find  the  exact  interest  at  4  %  on  a  note  for  $781.20,  dated 
June  5,  '89,  and  maturing  Oct.  4/7,  '92. 


$781.20 
.12 


Here  simple  interest  is  com- 
puted for  3  yr. ,  and  the  exact 


$  93.74  Interest  for  3  yr. 

9,00  Exact  interest  for  124  da.  mterest  for  124da-  (leaP  yr-)  1S 

$102.74  =  Ans. 


added. 


EXAMPLES  CV. 
Written  Exercises. 


What  is  the  amount  at  maturity  of  each  of  the  following 
indicated  notes,  exact  interest  ? 


Date.  Face. 

1.  Texas,  Jan.    4/93,  $890 

2.  K.  Y.,    Jan.    8/96,  $400 

3.  Mass.,  June  11,  '91,  $250 


Rate.  Maturity. 

41%  Apr.  4/7/93. 

4%  Apr.  7,  '96. 

6%  Mch.10/13,'92. 


4.  Oregon,  Apr.  19, '93,    $1250      Legal     July  18/21, '93. 

5.  Iowa,     Aug.  6,  '94,    $46849    Legal     Nov.  4/7,  '94. 

6.  Vt,       Feb.  4, '95,    $2685       5%       May  5, '95. 


Partial  Payments. 

251.  It  often  occurs  that  part  of  a  note  is  paid  at  one 
time,  another  part  at  another  time,  and  so  on,  until  all 
the  note  is  paid.  Such  payments  are  called  Partial 
Payments. 

In  case  of  interest-bearing  notes,  it  becomes  necessary 
to  compute  simple  interest  on  the  different  principals  which 
appear  during  the  life  of  the  note. 

The  sums  of  money  paid  and  the  times  of  payment  are 
Endorsed  on  the  back  of  the  note. 


270 


INTEREST. 


[Chap.  XI. 


Art.  251.] 


PARTIAL   PAYMENTS. 


271 


Dates  1 
the 

Jan. 

Mch. 
May 
July 
July 
Sept. 

$540. 

4.59 
$544.59 
75. 

found 
note. 

16,' 
8,' 
13, 

9, 
21, 
19, 

1st 
int, 
am 
1st 

2d 
int 

am 

2d 

3d 
int 

am 
3d 

4th 
int, 

am 
4th 

5th 
int, 

on        Times  between 
successive  dates. 

E 5ida- 

,94 66" 

•*::::::2  - 

£ co" 

prin. 

,  for  51  da. 
1't  of  1st  prin. 

payment. 

$469.59 
5.17 

prin. 

,  for  66  da. 

$474.76 
80. 

't  of  2d  prin. 
payment.                { 

$394.76 
3.75 

i 
prin.                        ( 

for  57  da.              j 

$398.51 
100. 

't  of  3d  prin.          ] 
payment.               T 

$298.51 
.60 

< 

prin. 

,  for  12  da. 

$299.11 
150. 

t  of  4th  prin. 
payment. 

$149.11 
1.49 

.  prin. 

,  for  60  da. 

When  a  note  is  wholly  within 
a  year,  the  exact  number  of  days 
between  dates  is  found,  and  the 
days'  method  is  used  in  compu- 
tation. 


It  will  be  observed  that  the 
amount  of  the  principal  is  found 
for  the  time  elapsing  between  the 
date  of  the  note  and  the  date  of  the 
first  payment.  The  first  payment 
is  subtracted,  and  the  remainder  is 
used  as  a  new  principal.  And  so 
on  to  the  end. 


$151.60    am't  paid  Sept.  19. 


272  INTEREST.  [Chap.  XI. 

Ex.2. 


&-.  0(yviria,jiw-LU,  Tttcl.,  f^}t&  /<?,  1888. 

Hn  cL&vwcwuL  J promise  to  pay  to 

the  order  of^^^^^/if&nvu  fo-kn^o-ru^^^^^^, 

wvwwwvww^  tkawQsa>nd ^Dollars 

at tk&  dfiaAsWv&Ws'  cfiaXAs(yyial>  Bank^^^^. 

Value  received,  w-vth  (mt&ve^C  at  ¥-°lo. 

^awv'l  £a>6-{>ttt. 


This  note  carried  the  following  endorsements : 

Dec.     1,  '88,  $150;  Mch.     1,  '92,  $1000; 

Apr.     7,  '89,  $250;  Mch.     1,  '93,  $2000. 

Oct.   25,  '90,  $275; 

Find  the  balance  which  was  paid  on  Sept.  19,  '94. 

Here  we  find  the  times  in  years,  months,  and  days. 

Dates  found  on  Times  between  Interest  on  $1  at  6% 

the  note.  successive  dates.  for  the  times, 

yr.  mo.  da. 

'88  6     19 5     12 $.027 

'88  12      X 4      6 $.021 

'89  4      7 16     18 $.093)       _„, 

;90  10    25 1     4      6 $.08ll=$'174 

'92  3      l 1              $.06 

'93      3      1 1    6     18 |.093 

'94      9     19 


Art.  252.]  U.  S.  RULE.  273 

$5000  1st  principal. 

90  int.  for  5  mo.  12  d.,  at  4%. 

$5090  am't  of  1st  prin. 

150  1st  payment. 

$4940.  2d  prin. 

69.16  int.  for  4  mo.  6  d.,  at  4%. 

$5009.16  am't  of  2d  prin. 

250.  2d  payment. 

$4759.16  3d  prin. 

552.06  int.  for  2yr.  10  mo.  24  d.,  at  4%. 

$5311.22  am't  of  3d  prin. 

1275.  3d  aDd  4th  payments. 

$4036.22  4th  prin. 

161.45  int.  for  1  yr.,  at  4%. 

$4197.67  am't  of  4th  prin. 

2000.  5th  payment. 

$2197.67  5th  prin. 

136.26  int.  for  1  yr.  6  mo.  18  da.,  at  4%. 

$2333.93  am't  paid  Sept.  19,  '94. 

In  case  any  payment  is  less  than  the  interest  due  at  the  time  of 
such  payment  (as  in  the  3d  payment  of  this  note)  a  portion  of 
the  interest  would  become  a  part  of  the  new  principal  and  would 
draw  interest,  if  we  should  proceed  as  with  the  1st  and  2d  pay- 
ments. Here  compound  interest  is  forbidden  by  law,  and  we  must 
find  the  interest  on  the  same  principal  until  the  time  when  the  sum 
of  the  payments  equals  or  exceeds  the  interest. 

The  United  States  Rule.* 

252.  Compute  the  amount  of  the  principal  to  the  time 
when  a  payment,  or  the  sum  of  two  or  more  payments, 
equals  or  exceeds  the  interest  due. 

Subtract  from  this  amount  the  payment,  or  the  sum  of 
the  payments,  and  proceed  with  the  remainder  as  a  new 
principal.     And  so  on  to  the  time  of  settlement. 

*  Vermont,  New  Hampshire,  and  Connecticut  have  methods  of 
their  own  for  computation  in  partial  payments,  but  it  is  not  advis- 
abje  to  consider  those  methods  in  our  present  study. 


274  INTEREST.  [Chaps.  XL,  XII. 

EXAMPLES   CVI. 

1.  A  Kentucky  note  for  $3500,  with  interest,  dated 
Mch.  1,  '90,  had  the  following  endorsements  : 

Apr.  6,  '90,  $500.  May  15,  '90,  $800. 

"  30,  '90,  $300.  July  11,  '90,  $600. 

What  was  paid  in  settlement  on  Aug.  22,  '90  ? 

2.  An  Arizona  note  for  $8600,  with  interest,  dated 
July  1,  '87,  had  the  following  endorsements : 

Oct.  2,  '87,  $150.  Feb.  21,  '88,  $4000. 

Nov.  7, '87,  $1500. 
What  was  due  May  4,  1888  ? 

3.  A  Louisiana  note  for  $876,   with  interest,  dated 
Feb.  6,  '86,  was  endorsed  as  follows : 

Apr.  11,  '86,  $50.  June  2,  '87,  $300. 

Dec.    1,  '86,  $150.  July  5,  '87,  $75. 

What  was  paid  in  settlement  on  Jan.  1,  '88  ? 

4.  A  Massachusetts  note  for  $3000,  with  interest  at 
4|%,  dated  Jan.  1,  '91,  was  endorsed  as  follows: 

Mch.   7,  '91,  $175.  Sept.  20,  '93, 

May    9,  '91,  $300.  Nov.  30,  '94, 

Aug.  17,  '93,  $400. 
What  was  paid  in  settlement  on  Dec.  5,  '94  ? 

5.  An  Indiana  note  for  $2500,  dated  Jan.  6,  '94,  was 
endorsed  as  follows: 


Feb.    7,  '94,  $250.  Oct.  6,  '94, 

Apr.  20,  '94,  $180.  Feb.  7,  '95,  $350. 

July   7,  '94,  $75. 
What  was  paid  in  settlement  on  Feb.  20,  '95  ? 

Ans.  $1145. 


Art.  253.]  DRAFTS.  275 


CHAPTER   XII. 

EXCHANGE. 

Drafts. 

253.  Suppose  that  Wilson  &  Co.  of  Baltimore  buy  of 
Morton  &  Co.  of  St.  Paul  $2500  worth  of  goods  on  60 
da.  credit.  When  the  bill  is  due,  Morton  &  Co.  may 
make  a  formal  request  for  its  payment.  Such  a  request 
is  called  a  Draft ;  Morton  &  Co.  are  said  to  draw  on 
Wilson  &  Co.     Por  example : 


#2500™.         £ft  JW,  TnUn.,  fidy  25,  18  <?f. 
^^^^^^^^^^^^JZt  Qyiakt^^^JPay  to  the 

Order  of^^^^^^^@wi&&lv-&Qs w^ 

£Tw-&riLu-lLv-&  AwyioOb&ci^ ^  Dollars 


WITH    EXCHANGE. 


Value  received  and  charge  the  same  to 
account  of 

to     fc>-&,       |    m<nto»*eo 

No.  2<f.       £altvnvov&,  Tfld.  J 


The  draft  is  sent  to  Wilson  &  Co.  through  a  St.  Paul 
bank  which  transmits  it  to  a  Baltimore  bank.  The  latter 
presents  the  draft  to  Wilson  &  Co.  for  payment,  and  the 
cash  is  sent  to  Morton  &  Co.  through  the  St.  Paul  bank. 

The  banks  charge  a  small  fee  for  their  services,  and  the  words 
'  with  exchange '  in  the  draft  signify  that  the  debtor  must  pay 
the  fee. 


276  EXCHANGE.  [Chap.  XII. 

The  above  draft  is  called  a  Sight  Draft. 

A  sight  draft  is  payable  on  presentation  (most  States 
not  allowing  grace  on  sight  drafts),  and,  from  its  nature, 
is  not  subject  to  discount. 

254.  Instead  of  waiting  for  the  expiration  of  the  60 
da.  and  then  drawing  'at  sight/  Morton  &  Co.  might 
make  a  Time  Draft,  payable  after  date.     For  example : 


$2500™ .  it.  JW,  Mm ,  fwUf  25,  1 8  <?¥. 

&cybty-^iA)~&  cLa,if&  ap^&v  dat&^^^Pay  to  the 

Order  o/_____w____fc^^^^&^^^^^^^^^^^ 

^^^^&w~emty-lAj$}  k^yicLv&cL .^  Dollars 

Value  reeeima  q$m  charge  the  same  to 
^    account  of 

To  TMUonV®*.,         )       motion  V  &o- 


No.  <?/6.    BMvywi*,  THcL-  J     tfefoL.  8/f/,  'w. 


If  Wilson  &  Co.  accede  to  the  request,  they  make  a 
formal  acceptance  of  the  draft  by  writing  across  its  face 
the  word  *  accepted,'  together  with  their  signature.  Their 
acceptance  is  equivalent  to  their  making  &  promissory  note, 
and  the  draft  is  regarded  as  such  by  all  concerned.  After 
acceptance,  the  draft  is  returned  (through  the  banks)  to 
Morton  &  Co.,  who  now  have  a  written  promise  from 
Wilson  &  Co.,  whereas  before  they  had  only  a  verbal 
promise.  Morton  &  Co.  now  have  the  draft  discounted, 
exactly  as  if  it  were  a  promissory  note,  and  thus  obtain 
the  cash  needed. 

255.  In  case  the  time  draft  is  made  payable  ' after  sight' 
instead  of  ' after  date'  Wilson  &  Co.  affix  to  their  accept- 
ance the  date  of  acceptance  so  that  maturity  may  be  found. 


Arts.  254,  255.]  DRAFTS.  277 

The  payee  is  the  owner  of  the   draft.     [See   also  Art. 
228,  Note.]     For  example  : 


f2500°-°-.  oft.  gcuul,  Mi™*,.,  fulif  25,  18  w. 

<^aiCu-tvv-&  cLaAffr  a^t&i>  QAsCfkt^^^JPay  to  the 

Order  o/^^^^^^^^^^^^^(5Z^^^^^^^^^^^^^ 

^^^^^w-e^tMh^y^Sf  havn^eds^^^^—lJollars 

Value  receid^a  arm  charge  the  same  to 
account  of 

To         TMUo-n,  V  €*.,  1       mavtaru  V  &*. 


,mci\ 


No.  <?/6.     8oJXJLwlo>u,,  md.)    tfefit.  W//3,  'W. 


Ex.  This  draft  was  discounted  at  6  %  on  July  29th ;  find  ma- 
turity and  proceeds. 

Mtm.       .8333 

fl  I  '  jjg         23  From  day  of  discount  to  maturity 

$  19.17  =  discount.  was  46  da. 

$  2480.83  =  proceeds. 

Time  drafts  are  rarely  used,  while  sight  drafts  are  very 
common. 

EXAMPLES   CVII. 

1.  E.  A.  Winslow  of  Brattleboro,  Vt.,  drew  on  F.  B. 
Crane  of  St.  Louis,  Mo.,  for  the  payment  of  a  $650  debt 
contracted  Apr.  13,  '92,  and  due  in  90  da.  The  draft  was 
dated  May  13,  '92,  and  made  payable  '  after  date.' 

Write  the  draft,  indicating  acceptance,  and  write  its 
date  of  maturity  in  the  lower  right-hand  corner. 

2 .  Eewrite  the  draft,  making  it  payable  'after  sight'  and 
find  its  maturity,  it  having  been  accepted  on  May  16,  '92. 

3.  Winslow  had  the  first  draft  discounted  May  20; 
find  the  proceeds. 


278  EXCHANGE.  [Chap.  XII. 

4.  What  would  have  been  the  proceeds  of  the  second 
draft,  had  it  been  discounted  May  18,  '92  ? 

5.  On  Jan.  1,  '92,  S.  B.  Titus  of  Austin,  Texas,  drew 
on  Ward  &  Co.  of  Macon,  Ga.,  for  the  payment  of  a 
$1765  debt,  contracted  Dec.  7,  '91,  and  due  in  90  da. 

Write  the  draft,  payable  '  after  sight,'  indicate  accept- 
ance on  Jan.  3,  '92,  and  write  its  date  of  maturity. 

6.  Draft  in  Ex.  5  was  discounted  Jan.  6,  '92;  pro- 
ceeds =  ? 

256.  It  is  evident  that  all  the  drafts  thus  far  shown 
have  been  requests  made  by  a  creditor  to  his  debtor. 
Now  drafts  may  be  used  for  paying  debts  as  well  as  for 
collecting  debts.  In  this  case  the  debtor  (through  his 
bank)  makes  a  draft  on  some  bank  in  the  city  where  his 
creditor  lives  and  payable  to  such  creditor. 


Domestic  Exchange. 

257.  The  main  object  of  drafts  is  the  payment  of 
debts  without  sending  the  actual  money,  thus  avoiding 
expense,  and  risk  of  loss. 

The  draft  method  of  making  payments  between  cities 
in  the  same  country  is  called  Domestic  Exchange, 

Foreign  Exchange. 

258.  The  draft  method  of  making  payments  between 
cities  in  different  countries  is  called  Foreign  Exchange. 

259.  Foreign  drafts  are  made  more  extended  in  form 
than  domestic  drafts,  and  are  called  Bills  of  Exchange. 
A  Bill  of  Exchange  consists  of  a  set  of  two  bills,  both  alike, 


Arts.  256-262.]         FOREIGN  EXCHANGE.  279 

except  that  they  are  numbered.  These  two  bills  are  sent 
by  different  steamers,  and  as  soon  as  one  of  the  bills  has 
been  paid  the  other  becomes  void. 

260.  The  drawing  of  Bills  of  Exchange  is  done  by 
brokers,  and  no  commission  is  charged  for  transacting 
the  business. 

261.  The  actual  amount  paid  for  Bills  of  Exchange, 
for  example  paid  in  New  York  for  bills  on  London,  varies 
from  time  to  time ;  the  current  price  paid  for  Bills,  called 
the  'Rate  of  Exchange/  cannot,  however,  ordinarily  be 
much  above  or  below  par ;  for  if  it  would  cost  more  to 
discharge  a  debt  by  means  of  a  bill  than  by  the  actual 
transmission  of  bullion,  the  latter  method  would  naturally 
be  adopted. 

It  should  be  noticed  that  even  if  all  countries  had 
exactly  the  same  coinage,  there  would  still  be  fluctuations 
in  the  rate  of  exchange  between  two  countries,  as  the 
balance  of  indebtedness  between  those  two  countries 
varied. 

262.  The  following  table  gives  the  value  of  some 
foreign  coins  in  terms  of  U.  S.  Money  as  proclaimed  by 
the  Secretary  of  the  Treasury  on  Jan.  1,  '95 : 

Austria 1  Crown  =  $  .20,  3 

Belgium 1  Franc  =  .19,  3 

Brazil 1  Milreis  =  .54,  6 

Chili 1  Peso  =  .91,  2 

,  m    ,  r  Shanghai  =      .67,  3 

China 1  TaeW  TT  ..  &  ' 

I  Haikwan  =      .74,  9 

Cuba 1  Peso  =      .92,  6 

France 1  Franc  =      .19,  3 

Germany 1  Mark  =      .23,  8 

Great  Britain    ....  1  Pound  Sterling  =  4.86,  Q\ 

Holland 1  Guilder  =      .40,  2 

Italy 1  Lira  =      .19, 3 


280 


EXCHANGE. 


"hap. 

XII. 

.99, 

7 

.98, 

8 

.26, 

8 

■77, 

2 

.19, 

3 

.26, 

8 

.19, 

3 

Japan 1  Yen  (gold) 

Mexico 1  Dollar  (gold) 

Norway 1  Crown 

Russia 1  Rouble  (gold) 

Spain 1  Peseta 

Sweden 1  Crown 

Switzerland 1  Franc 

These  values  are  subject  to  change. 


263.  Exchange  on  Great  Britain  is  quoted  at  the  value 
of  one  pound  sterling  (£  1)  in  dollars;  exchange  on  France 
is  quoted  at  the  number  of  francs  to  the  dollar;  exchange 
on  Germany  is  quoted  at  the  value  of  four  reichsmarks. 

The  following  is  copied  from  a  daily  journal  : 

The  foreign  exchange  market  was  steady,  but  very  quiet  in 
tone.  Posted  rates  were  unchanged  at  $4.88|  for  sixty-day  bills 
and  $4.90  for  demand.  Actual  sales  were  $4.87§  @  .$4.88  for 
sixty-day  bills,  $4.89^  for  demand,  $4.89£  for  cables,  and  $4.87  @ 
$4.87^  for  commercial. 

In  Continental,  francs  5. 17|  for  long  and  5. 16 J  for  short ;  reichs- 
marks 95^  and  95| ;  guilders  at  40$  and  40f . 

The  following  example  shows  the  form  of  a  Bill  of 
Exchange  and  how  to  find  its  cost. 


£1200^-.  c/W  Ifcyyfo,  cA. If.,  fot.  2f,  18  <?$. 

ojf  th&  &wm>&  dat&  avid  t&ruyu,  wn/foaAxL 

^^^^^^^^v^^^^^  to  the 

Order  of ^cwyvw&l  Llttl&jahyi^^^^^^ 

Value  received  and  charge  the  same  to 
To  I  c/.  mo*****,  V  Go.,  ) 

JUyncucyyv,  (on^Canci       )  ' 


Art.  263.]  EXAMPLES.  281 

On  Oct.  29th  sight  drafts  on  London  were  quoted  at  $4.88|. 

£1200 
4.88£ 


$5862.00  =  cost  of  exchange. 

Ex.  2.   How  large  a  sight  draft  on  London  can  be  purchased 
for  $3890,  exchange  at  4.86£? 

4.86^)3890. 

£800.=^ns. 


EXAMPLES   CVIII. 

Find  the  cost  in  New  York  of  a  Bill  of  Exchange  for 

1.  £500  on  London  at  4.86^. 

2.  £1750  on  Glasgow  at  4.85. 

3.  50000  francs  on  Paris  at  5.18f . 

4.  1250  marks  on  Berlin  at  95 \. 

5.  2000  milreis  on  Rio  Janeiro  at  54.9  [cents  per 
milreis]. 

6.  3000  crowns  on  Vienna  at  par. 

7.  Calculate  the  cost  at  market    (  a.    £650. 
prices    (as  found    in    some    daily   <  b.   2400  francs, 
journal)  of  (  c.    2000  marks. 

8.  What  will  be  the  face  of  a  N.  Y.  draft  on  Bremen 
costing  $297.96,  exchange  being  at  95 \  ?  (Omit  decimals 
of  the  answer.) 

9.  How  large  a  draft  on  London  can  be  purchased  for 
$8554.14,  exchange  being  quoted  at  4.88J  ? 

10.   How  large  a  draft  on  Paris  can  be  purchased  for 
$1920,  exchange  being  quoted  at  5.18|? 


282  STOCKS  AND  BONDS.  [Chap.  XIII. 


CHAPTER  XIII. 

STOCKS  AND  BONDS. 
Stocks. 

264.  There  are  many  business  undertakings,  such  as 
railways,  banks,  gas  works,  etc.,  which  are  on  so  large  a 
scale  that  many  persons  must  combine  to  provide  the 
money  necessary  to  carry  on  the  business.  This  is  gen- 
erally done  by  dividing  up  the  whole  sum  required  into 
1  Shares '  of  definite  amount,  say  of  $10,  or  $50,  or  $100 
each. 

The  whole  body  of  partners  is  called  a  Company,  and 
the  individual  partners  are  called  Stockholders. 

The  total  amount  of  money  raised  to  carry  on  the  busi- 
ness of  the  company  is  called  its  Capital. 

The  affairs  of  a  company  are  managed  by  a  small 
number  of  elected  stockholders  called  Directors. 

The  profits  made  by  the  company  are  called  Dividends, 
and  are  periodically  divided  among  the  stockholders ;  the 
dividend  is  declared  as  a  percentage  on  the  capital. 

265.  A  stockholder  in  a  company  cannot  demand  the 
return  of  the  money  he  paid  for  his  shares ;  he  can,  how- 
ever, sell  the  shares. 

If  the  dividends  of  the  company  are  high,  and  are 
likely  to  continue  to  be  high,  the  shares  will  sell  for 
more  than  they  originally  cost;   if,  however,  the   com- 


Arts.  264-268.]  PREMIUM  —  DISCOUNT.  288 

pany  is  not  prosperous,  the  shares  would  have  to  be  sold 
for  less  than  they  originally  cost. 

Thus,  the  stockholders  in  a  company  are  continually 
changing,  and  different  stockholders  may  have  bought 
their  shares  at  very  different  prices. 

266.  The  most  important  point  to  notice  is  that  the 
amount  of  dividend  paid  to  a  stockholder  does  not  depend 
on  the  price  at  which  his  shares  were  bought,  but  simply  on 
their  nominal  value. 

Thus,  two  men  who  had  the  same  number  of  $100  shares  in  a 
company  would  be  entitled  to  the  same  amount  of  dividend, 
although  one  may  have  bought,  for  example,  $100  shares  for  $180 
and  the  other  for  $50  each. 

267.  Shares  are  said  to  be  above  or  below  'par'  ac- 
cording as  they  are  sold  for  more  or  for  less  than  their 
nominal  value.  The  nominal  value  is  $  100  per  share, 
unless  otherwise  stated. 

Thus,  if  $100  shares  sell  for  $110  each,  since  $110  is  \%%  of 
$100,  the  shares  are  10  per  cent  above  par. 

When  the  price  of  shares  is  more  than  their  nominal 
value  they  are  said  to  be  *  at  a  premium, ■  and  when  the 
price  is  less  than  their  nominal  value  the  shares  are  ■  at 
a  discount.' 

268.  The  following  are  examples  of  the  different  ques- 
tions which  may  have  to  be  considered. 

Ex.  1.  $100  shares  in  a  gas  company  sell  for  $240  each;  how 
much  will  70  shares  cost  ? 

Each  $100  share  costs  $240  cash ; 

.  •.  70  shares  cost  $240  x  70  =  $16800. 

Ex.  2.  A  man  bought  $100  shares  in  a  gas  company  for  $16800, 
giving  $240  for  each  $100  share  ;  how  many  shares  did  he  buy  f 
Since  each  share  cost  $240, 

the  number  of  shares  =  $16800  -r-  $240  =  70. 


284  STOCKS  AND   BONDS.  [Chap.  XIII. 

Ex.  3.  A  gas  company  pays  a  dividend  of  8%  per  annum;  how 
much  does  a  man  receive  who  holds  70  $100  shares  ? 

His  share  of  the  capital  is  $100  x  70  =  $7000,  and  he  receives 
8%  on  this,  or  $560. 

Ex.  4.  A  man  invests  money  in  the  stock  of  a  company,  each 
$100  share  costing  $240 ;  what  %  does  he  receive  on  his  investment 
when  the  company  pays  an  8%  dividend  ? 

He  receives  $8  on  each  share,  and  having  paid  $240  for  a  share, 
he  receives  $8  on  each  $240  invested ;  ^|¥  =  3£%. 

269.  Sometimes  a  company  does  not  need  its  full 
capital  to  carry  on  its  business ;  and  in  that  case  only  a 
certain  fraction  of  the  nominal  amount  of  the  shares  is 
'  paid  up ' ;  the  stockholders  are,  however,  bound  to  pay 
the  rest  if  it  should  become  necessary.  When  a  dividend 
is  declared  at  so  much  per  cent,  this  percentage  is  paid 
only  on  the  amount  paid  up  on  the  shares,  and  not  on 
their  full  nominal  value. 

Ex.  What  income  will  be  obtained  by  investing  £1008  in  the 
purchase  of  £20  bank  shares,  on  each  of  which  £5  is  paid  up,  at 
£24  each  share,  the  bank  paying  a  dividend  of  18  per  cent  ? 

Since  £24  buys  one  share,  £1008  will  buy  £1008  --  £24  =  42 
shares.  These  42  shares,  on  each  of  which  £5  is  paid,  make  up  a 
capital  of  £5  x  42  =  £210.  On  this  capital  of  £210  a  dividend  of 
18%  is  paid ;  hence,  income  required  =  £210  x  TVo  =  £37.  16s. 


EXAMPLES  CIX. 
Written  Exercises. 

1.  If  $10  shares  sell  for  $3.50,  how  many  shares  can 
be  bought  for  $9271.50  ?  What  is  the  nominal  value  of 
shares  purchased  ? 

2.  Mining  shares  of  $10  each  are  sold  at  $2.50  dis- 
count ;  what  is  the  price  of  80  shares  ? 


Art.  269.]  EXAMPLES.  •  285 

3.  The  shares  of  a  certain  company  are  sold  at  10% 
above  par ;  how  much  must  be  paid  for  1060  $50  shares  ? 

4.  A  company  pays  a  dividend  of  8%;  how  much 
does  A  receive  if  he  holds  50  $50  shares  ? 

5.  A  man  holds  350  shares  of  $50  each,  and  the 
company  pays  7%  dividend;  how  much  does  he  receive? 

6.  A  man  sells  63  $100  shares  for  $180  each,  and 
buys  with  the  proceeds  $50  shares  at  $35  each;  how 
many  shares  does  he  buy  ? 

7.  What  is  the  difference  between  a  $100  stock  and 
$100  worth  of  stock? 

8.  A  man  sold  75  $50  shares  for  $65  each,  and  in- 
vested the  money  in  $100  shares  at  $125  each;  how 
many  shares  did  he  buy  ? 

9.  What  income  would  be  obtained  by  investing 
$3850  in  the  purchase  of  $100  shares  in  a  company  at 
$175  each,  the  company  paying  a  dividend  of  6%  per 
annum  ? 

10.  $100  shares  in  a  certain  bank  sell  at  $350,  and 
the  bank  pays  a  semi-annual  dividend  of  7%  ;  what 
annual  income  would  be  obtained  by  investing  $9450  ? 

11.  A  company  pays  a  dividend  of  4j-%,  and  its  $100 
shares  sell  for  50%  above  par;  what  per  cent  does  an 
investor  receive  ? 

12.  A  man  buys  $50  shares  at  $62.50,  and  the 
company  pays  a  5%  dividend;  what  percentage  does  he 
receive,  and  what  %  on  his  investment  ? 

13.  A  man  sells  fifty  shares  of  $100  gas  stock,  paying 
8%  dividend,  at  $180;  he  invests  the  proceeds  in  $50 
railway  stock  at  $35 ;  find  the  change  in  his  income,  the 
railway  company  paying  a  dividend  of  3^-%. 


286  STOCKS  AND  BONDS.  [Chap.  XIII. 

14.  A  man  buys  $100  stock  in  a  company  which  pays 
an  8  %  dividend,  and  he  buys  at  such  a  price  as  to  receive 
3%  on  his  investment;  what  does  he  pay  per  share? 

15.  A  bank  pays  a  9%  dividend,  and  its  $600  shares, 
of  which  $200  is  paid  up,  sell  for  $750 ;  what  %  does 
an  investor  receive  on  his  money  ? 

The  price  of  Stock  is  given  at  so  much  per  cent;  thus, 
stock  is  said  to  be  at  115,  when  $100  stock  costs  $115, 
and  so  in  proportion  for  other  amounts. 

16.  How  much  will  $500  stock  at  75  sell  for? 
How  much  will  $150  stock  at  120  sell  for? 
How  much  will  $60  stock  at  128  sell  for  ? 
How  much  will  $1200  stock  at  97  sell  for? 

17.  What  income  will  be  obtained  from  $500  stock 
when  the  dividend  is  4%  ? 

18.  What  income  will  be  obtained  by  investing 
$110175  in  a  stock  which  pays  3^-%,  and  can  be  bought 
at  113? 

19.  What  income  will  be  obtained  by  investing  $70380 
in  a  31%  stock  at  97f? 

20.  What  °]0  will  a  man  get  on  his  money  if  he  invests 
in  a  4%  stock  at  125? 

21.  A  man  receives  $660  a  year  by  investing  $21450 
in  4%  railway  stock;  what  was  the  nominal  value  of  the 
stock? 

22.  An  income  of  $506.25  per  year  is  derived  by 
investing  $15300  in  a  4$-%  stock ;  what  was  the  price  of 
the  stock  per  share  ? 

23.  Stock  was  purchased  at  97-J-  and  sold  at  103 J,  and 
the  profit  was  $661.25 ;  how  much  stock  was  purchased 
and  what  was  the  total  cost  ? 


Arts.  270-272.]  BONDS.  287 

24.  In  which  will  a  man  receive. the  greater  %  on  his 
investment ;  in  a  3%  stock  at  95  or  in  a  4%  at  127  ? 

25.  What  will  be  the  difference  in  income  between  a 
4%  stock  at  129  and  a  4i%  at  145  ? 

Bonds. 

270.  Governments  borrow  money  to  meet  exceptional 
expenditure,  and  undertake  to  pay  a  fixed  rate  of  interest. 

The  promissory  notes  given  in  return  for  this  money 
are  called  Bonds.  The  bonds  differ,  however,  from  the 
ordinary  promissory  notes  in  being  more  formal,  and  in 
having  small  certificates  attached  to  enable  the  holder  of 
the  bond  to  easily  collect  his  interest.  These  certificates 
are  called  Coupons.  There  is  a  coupon  for  each  3  mo. 
of  interest.  Therefore  a  twenty-year  bond  has  eighty 
coupons  attached. 

Eailway  and  other  companies  generally  issue  bonds  of 
a  nature  similar  to  that  of  government  bonds. 

271.  A  person  investing  money  in  bonds  is  sure  of 
a  specified  income,  while  a  person  investing  in  stocks 
receives  only  his  share  of  the  profits  after  all  expenses, 
including  the  interest  on  bonds,  have  been  paid. 

272.  The  public  debt  of  the  United  States  Apr.  1,  '95. 


Amount  of  Bonds. 

Rate. 

When  Redeemable. 

$25,364,500 

2%. 

Option  of  U.S. 

559,624,850 

4" 

July  1,  1907. 

54,710 

4" 

100,000,000 

5" 

Feb.  1,  1904. 

28,807,900 

4" 

Feb.  1,  1925. 

$713,851,960.00  Total  int.-bearing  debt. 
381,025,096.92  Non  int.-bearing  debt  (U.S.  Notes,  Nat.  Bank 
Notes,  Fractional  Currency). 
1,770,250.26  Debt  which  has  matured. 


$1,096,647,307.18  Total  debt,  exclusive  of  bonds  issued  to  Pacific 
railroads. 


288  STOCKS  AND   BONDS.  [Chap.  XIII. 

273.  Stocks  and  bonds,  except  those  of  small  com- 
panies, are  bought  and  sold  at  a  special  market,  called  a 
Stock  Exchange.  The  agent  who  is  employed  to  buy  and 
sell  for  the  public  is  called  a  Stock  Broker,  and  the  person 
who  deals  in  stocks  and  bonds  is  called  a  Stock  Jobber. 

Stock  Brokers  charge  for  their  services  a  commission 
called  Brokerage;  in  calculating  the  cost  of  stocks  and 
bonds  this  brokerage  must  be  added  to  their  market 
prices;  the  proceeds  of  a  sale  of  stocks  and  bonds  are 
the  market  prices  minus  the  brokerage.  In  previous 
examples,  brokerage  has  been  allowed  for  in  the  prices. 

274.  Brokerage  is  generally  \  of  1%,  reckoned  on  the 
par  value  of  the  stock ;  it  is  therefore  £  of  $1  on  every 
$100  share  bought  or  sold,  no  matter  what  the  market 
price. 

(In  the  following  examples  each  share  is  to  be  considered  as 
§  100  par  value,  and  }%  is  to  be  allowed  for  brokerage.) 

Ex.  A  man  sold  out  $5000  stock  of  a  company  which  paid  3|% 
annual  dividends  at  94},  and  invested  the  proceeds  in  a  stock  which 
paid  4%  at  108  \  ;  what  was  his  change  in  income  ? 

$5000       x  M\      =  $175         =  original  income. 

91}        —  }  =  $91}        =  proceeds  from  one  share. 

$91}        x  50         =  $4556.25  =  proceeds  from  50  shares. 

108}       +  }  =  $101}       =  cost  of  each  new  share. 

$4556.25  h-  101.25  =  45  =  number  of  new  shares. 

$4500       x  .04       =  $180      .  =  income  from  new  shares. 
.  \  he  had  an  increase  of  $5  in  his  income. 

EXAMPLES   CX. 

1.  What  is  the  difference  between  a  dollar  of  stock 
and  a  dollar's  worth  of  stock  ? 

2.  What  is  the  difference  in  the  interests  on  a  hun- 
dred-dollar stock  and  a  hundred-dollar  bond  ? 


Arts.  273,  274.]  EXAMPLES.  ,  289 

3.  What  amount  of  bonds  at  97f  can  be  bought  for 
$3900  ? 

4t  What  amount  of  bonds  at  96$  can  be  bought  for 
$5335? 

5.  What  number  of  bonds  at  97J  can  be  bought  for 
$7154  ? 

6.  What  number  of  bonds  at  97^  can  be  bought  for 

$584.25? 

How  much  would  "be  realized  by  selling 

7.  $1000  bonds  at  96? 

8.  $500  bonds  at  981  ? 

9.  $100  bonds  at  118$? 

10.  Bonds  bought  at  124|  pay  5%  on  the  investment; 
what  rate  do  they  bear  ? 

11.  Bonds  bought  at  92$  pay  4$f  %  on  the  investment ; 
what  rate  do  they  bear  ? 

12.  What  is  the  price  of  U.S.  5  per  cents  when  the 
investment  produces  4T6T%  ? 

13.  I  have  $10000  to  invest  in  U.S.  4's  at  118$;  what 
is  my  income,  and  how  much  money  is  not  invested  ? 

14.  I  have  $7000  to  invest  in  U.S.2's  at  107f ;  what 
is  my  income,  and  how  much  money  remains  uninvested  ? 

15.  U.S.  2's  are  bought  at  114$;  what  rate  do  they 
bear? 

16.  The  trustees  of  a  school  invested,  as  a  teachers' 
fund,  $40512.50  in  U.S.  5's  at  115f ;  the  salary  of  the 
principal  was  $1000 ;  how  much  was  left  for  his  assistant  ? 

17.  A  speculator  invested  in  a  company  and  received 
a  dividend  of  6%,  which  was  8|%  on  the  investment;  at 
what  price  did  he  purchase  ? 


290  STOCKS  AND   BONDS.     [Chaps.  Xllt,  XIV. 

18.  A  young  man  receiving  a  legacy  of  $48000  invested 
one  half  in  5%  railway  bonds  at  95J,  and  the  other  half 
in  6%  stock  at  119 J;  what  income  did  he  secure  ? 

19.  A  owns  a  farm  which  rents  for  $320.40  per  yr. 
If  he  should  sell  the  farm  for  $8010  and  invest  the  pro- 
ceeds in  U.S.  4's  at  111^,  will  his  yearly  income  be 
increased  or  diminished,  and  how  much  ? 

20.  A  capitalist  drew  the  quarterly  interest  on  his 
U.S.  4's,  amounting  to  $540,  and  afterwards  sold  the 
bonds  at  $124§;  what  were  the  proceeds  of  the  sale  ? 

21.  A  lady  invested  $20948.75  as  follows:  $6160  in 
Maryland  6's  at  96 \,  $8225  in  manufacturing  stock  at 
87f  paying  8%  annual  dividends,  and  the  remainder  in 
steamboat  stock  at  73f  paying  10%  annual  dividends; 
what  was  her  total  income  ? 

English  government  bonds  are  called  Consols. 

22.  A  man  had  £2400  in  the  2|%  consols;  he  sold 
out  at  99  J-  and  invested  the  proceeds  in  4%  railway  bonds, 
thereby  increasing  his  income  by  £6  ayr;  at  what  price 
did  he  buy  the  bonds  ? 

23.  A  man  having  an  income  of  £352  a  yr.  in  the 
2f  %  consols,  sells  out  at  97  and  invests  the  proceeds  in 
4%  railway  bonds,  thereby  increasing  his  income  £48  a 
yr ;  at  what  price  were  the  bonds  purchased  ? 


Abts.  275-277.]      ARITHMETICAL  PROGRESSION.  291 


CHAPTER   XIV. 

PROGRESSIONS. 

275.  A  series  of  numbers  which  increases  or  decreases 
regularly  is  called  a  Progression. 

For  instance,  3,   5,    7,    9,    11,  or  23,  20,  17,  14,  11,  8, 

or  3,  6,  12,  24,  or  81,  27,    9,    3,    1,    |,  *, 
are  progressions. 

It  will  be  noticed  that  in  the  first  two  progressions  the  series  are 
made  by  successive  additions  or  subtractions,  while  in  the  last  two 
the  series  are  made  by  successive  multiplications  or  divisions. 

The  first  are  called  Arithmetical  Progressions  (increasing 
or  decreasing). 

The  second  are  called  Geometrical  Progressions  (increas- 
ing or  decreasing). 

Arithmetical  Progressions. 


276. 

There  are  five  things  to  be  considered 

the  first  term, 

denoted 

by 

a, 

the  last  term, 

« 

u 

I, 

the  number  of  terms, 

u 

ii 

n, 

the  common  difference, 

a 

ft 

d, 

and  the  sum  of  the  terms,       "         "   s. 

277.  Any  three  of  these  five  being  given,  the  other  two 
may  be  found. 


292  PROGRESSIONS.  [Chap.  XIV. 

In  the  arithmetical  progression, 

7,  10,  13,  16,  19,  22,  25, 

it  is  evident  that  the  last  term  is  a  plus  six  d,  or  that  the  first  term 
is  I  minus  six  d. 

.•.  I  =  a  +  (n  —  1)  d, 

and  a  =  I  —  (n  —  1)  d. 

It  is  also  evident  that  if  a  and  I  be  added  and  the  sum  -f-  2,  the 
result  will  be  the  middle  term  ;  and  that  if  each  term  be  changed 
so  as  to  contain  as  many  units  as  the  middle  term  the  sum  of  the 
new  series  will  be  the  same  as  the  sum  of  the  original  series. 

2 

By  these  formulas  all  examples  in  arithmetical  progression  may 
be  solved. 

Ex.  1.  a  =  3,  d  =  5,  n  =  12  ;  find  I  and  s. 

Now   I  =  a  +  (n  —  1)  d  _  g  -j-  I 

=  3  +  11x5  S~     2 

=  58.  =  3  +  58  x  12 

=  30.5  x  12 


Ex.  2.   a  =  5, 1  =  17,  n  =  7  ;  find  d. 


Now 

1  —  a+  (n  —  1)  d; 

.-.     17  =  5  +  6d; 

whence 

6d  =  12, 

and 

d=    2. 

Ex.  3. 

Fmd  n  when  a  =  2,  Z  =  30,  and  d  =  7, 

Now 

Z  =  a  +  (n  —  1)  d; 

.-.    30  =  2  +  0-1)7; 

whence 

7  (»  -  1)  =  28, 

and 

w-l  =  4; 

i.e.. 

n  =  5. 

Art.  278.]         GEOMETRICAL   PROGRESSION.  293 


EXAMPLES   CXI. 

Written  Exercises. 

Answer  the  indicated  questions. 

1. 

2. 

3.                  4. 

5. 

6. 

a=       12. 

5. 

1.                   ? 

? 

.24. 

1  =        ? 

41. 

4.5.            351 

18. 

? 

d=        5. 

4. 

?               3|. 

3. 

1.2. 

n=        8. 

? 

8.              6. 

6. 

7. 

s  = 

9 

?                ? 

? 

7.    Insert  3 

means  between  2  and  12. 

8.    Find  the 

i  series  of  8  terms  when  the  3d  term 

is  14 

and  the  7th  term  is  26. 

9.   Find  the  series  of  9  terms  when  a  =  10.8  and  the 
6th  term  =  4.8. 

10.  Find  2  +  5  +  8  +  11  +  ...  to  37  terms. 

11 .  Find  8  +  7.75  +  7.5  H to  11  terms. 

Geometrical  Progressions. 
278.   There  are  five  things  to  be  considered : 
the  first  term,  denoted  by  a, 


the  last  term, 

u 

"     I, 

the  number  of  terms, 

u 

«    n, 

the  ratio, 

a 

the  sum  of  the  terms, 

a 

"     S. 

(The  ratio  is  the  relation  existing  between  any  two  successive 
terms.  It  is  the  constant  multiplier  by  which  any  term  is  found 
from  the  preceding  term. ) 

Any  three  of  these  five  being  given,  the  other  two  may 
be  found. 


294  PROGRESSIONS.  [Chap.  XIV. 

In  the  geometrical  progression, 

2,  6,  18,  54,  162, 

it  is  evident  that  the  last  term  is  a  times  the  product  of  r  by  itself 
four  times,  i.e.,  a  x  r4. 

Ia  l  =  ?  *  Z "M formula  1. 
and  a  =  Z  ■+•  r»  - l.  J 

It  is  also  evident  that 

8  =  2  +  6  +  18  +  54  +  162  ;  (1) 

multiplying  the  equation  by  the  ratio, 

3s  =  6+ 18  +  54  +  162  +  486;  (2) 

subtracting  (1)  from  (2),  we  have 

3s -s  =486-2, 
or  s(3-l)  =  486-2; 

whence  .      486-2 


3-1 
Now  486  =  rl,  2  =  a,  and  3  =  r  ; 

=  rj^-a,    formula  2. 
r-  1 

By  means  of  these  two  formulas  all  examples  in  geometrical 
progression  may  be  solved. 


Ex.  1. 

a  =  3,  r  =  2, 

n  = 

-  5  ;  find  I  and  s. 

Now 

1  =  arnl 
=  3x2* 
=  48. 

s  =  rZ~a 
r-  1 

2  x  48  -  3 

2-1 

=  93. 

Ex.2. 

a  -  3,  I  =  81 

i  n 

=  4  ;  ^nd  d. 

Now 

1  =  ar71-1  ; 

whence 

81  =  3  x  r3  : 

whence 

r8  =  27; 

whence 

r  =  3. 

Art.  279.]  INFINITE   SERIES.  295 


Ex.  3. 

Find 

n 

when 

a  =  3,  I 

=  375, 

and  r  - 

=  5, 

Now 

1  = 

arn  - 1 

> 

whence 

375  = 

3x5" 

-l . 

whence 

125  = 

5*-1; 

whence 

n-l  = 

3; 

whence 
Formu 

ila  2  ' 

be 

icomej 

n  = 

:4. 

~rlM 

(2)  is  i 

5llb 

if  (2)  is  subtracted  from  (1). 
1  —  r 

This  should  be  used  in  case  of  a  decreasing  geometrical  progression. 


EXAMPLES   CXII. 


Written  Exercises. 

Answer  the  indicated  questions. 

l. 

2. 

3. 

4. 

5. 

6. 

a=     2. 

11. 

1 
2"* 

? 

? 

1.3. 

1  =     ? 

352. 

625 

rsr- 

2744 

2  16  • 

608. 

? 

r  =     5. 

2. 

? 

f 

2. 

1.2. 

71=      5. 

? 

5. 

4. 

6. 

4. 

s  = 

9 

9 

7 

? 

7.  Insert  3  geometrical  means  between  4  and  2500. 

8.  Find  the  series  of  8  terms  when  the  3d  term  is  10.8 
and  the  7th  is  874.8. 

9.  Find  the  series  of  6  terms  when  a  =  yt  and  the 
fourth  term  is  J-ff|. 

10.  Find  2£  +  6f  +  19$  +  —  to  10  terms. 

11.  Find  the  series  of  5  terms  when  a  =  36.015  and 
the  3d  term  is  .735. 

12.  Find  28.8  + 14.4  + 7.2  +  ...  to  7  terms. 

279.  When  a  decreasing  geometrical  series  is  extended 
to  a  large  number  of  terms,  the  last  term  will  be  so  small 
that  it  will  have  no  appreciable  value. 


296  PROGRESSIONS.         [Chaps.  XIV.,  XV. 

Thus,  if  we  continue  f,  &  Tfa,  7|T,  „*„,  „£„,  indefinitely,  the 

last  term  will  be  almost  zero  ;  .-.in  the  formula  s  —  a~r  the  Ir 

1  -r 
of  the  numerator  may  be  omitted,  and  the  formula  will  become 

s  =     a    ,  by  which  we  may  find  the  sum  of  the  terms  of  a  decreas- 
1  —  r 


ing  infinite  series. 

Ex.1.    JP&Kg  $+ $ +  ^ +£  +  .., 

to  infinity. 

s « L 

--i- a 

S-l-r~l- 

i  —  i  —  f  • 

2            2 

Ex.  2.   jFmd  Me  value  of  .46. 

Now  .46  =  .4  +  .06  +  .006  +  .0006,  etc. 

.*.  the  value  must  equal  .4  +  the  geometrical  progression,  .06, 
.006,  .0006,  etc. 

.      -a        .06  _  . 

.-,  .46  =  .4  +  ^  =  H  +  A  =  A- 


EXAMPLES   CXIII. 
Written  Exercises. 

1.  Find  i  +  A  H to  infinity. 

2.  Find  ]  +  f  +  ...  to  infinity. 

3.  Find  the  value  of  1.416. 

4.  Find  the  value  of  1.53L 

5.  Find  the  value  of  3.3360. 


Arts.  280,281.]  CUBE    ROOT.  297 


CHAPTER  XV. 

CUBE   ROOT. 

280.  The  cubes  of  the  first  10  whole  numbers  should 
be  known :  they  are 

l;  8,  27,  64,  125,  216,  343,  512,  729,  1000. 

An  integer  (or  a  fraction)  which  is  the  cube  of  another 
integer  (or  fraction)  is  called  a  Perfect  Cube. 

Thus,  64  and  iff  are  perfect  cubes ;  namely,  the  cubes  of  4  and 
4  respectively. 

281.  In  simple  cases  the  cube  root  of  a  number  can 
be  found  by  separating  it  into  factors,  as  in  Art.  80. 

For  example,  to  find  ^/9261. 

9261  =  9  x  1029  =  27  x  343  =  3*  x  73  =  (3  x  7)8  ; 
hence,  ^/9261  =  f/(3  x  7)3  =  3  x  7  =  21. 

EXAMPLES  OXIV. 
Find  the  cube  root  of  each  of  the  following  numbers : 

1.  10648.  3.   35937.  5.   19683. 

2.  3375.  4.    13824.  6.   42875. 

Find  the  least  number  by  which  each  of  the  following 
numbers  must  be  multiplied  in  order  that  the  result  may 
be  a  perfect  cube. 

7.  108.  9.   336.  11.   4032. 

8.  392.  10.   441  12.    7056. 


298  CUBE   ROOT.  [Chap.  XV. 

282.  Since, 

103  =  1000,  1003  =  1000000,  10003  =  1000000000, 
and  so  on,  it  follows  that 

if  a  number  has  1  digit,  its  cube  has  either  1,  2,  or  3  digits 
"  "         2  digits,  "  «  4, 5,  or  6     " 

"  «         3      "       «  "  7,8,  or  9     " 

Hence,  if  we  mark  off  the  digits  of  a  given  number,  be- 
ginning at  the  units'  digit,  in  periods  of  three,  the  last  of 
the  periods  containing  one,  two,  or  three  digits ;  then  the 
number  of  these  periods  will  be  equal  to  the  number  of  digits 
in  the  cube  root  of  the  given  number. 

For  example,  by  pointing  off  the  numbers,  2744,  32.768,  3511808, 
as  follows,  namely,  2'744,  32'.768,  and  3'511'808,  we  see  that  the 
cube  roots  of  these  numbers  contain,  respectively,  2,  2,  and  3  figures. 

Find  (60  +  3)3. 

By  Art.  86,         (60  +  3)2  =  602+  2(60  x  3)  +  32 

Multiplying  by  60+3 

603+2(602x3)  +     60  x  32 

602x3  +2(60x32)  +  38 
and  (60  +  3)»  =  603+  3(602  x  3)  +  3(60  x  32)  +  3*. 

The  cube  of  the  sum  of  any  two  other  numbers  can  be 
expressed  in  a  similar  form. 

Hence,  the  cube  of  the  sum  of  any  two  numbers  is  equal 
to  the  cube  of  the  first  plus  three  times  the  square  of  the  first 
multiplied  by  the  second  plus  three  times  the  first  multiplied 
by  the  square  of  the  second  plus  the  cube  of  the  second. 

The  above  Theorem  will  enable  us  to  find  the  Cube 
Root  of  any  number. 

283.  To  find  the  Cube  Root  of  any  number.  The  method 
will  be  seen  from  the  following  examples: 


Arts.  282,  283.]  CUBE   ROOT.  299 

Ex.  1.    To  find  the  cube  root  of  157464. 

By  pointing  off  the  figures  into  periods  of  three  [Art.  282],  we 
see  that  there  are  two  figures  in  the  required  root. 

The  first  figure  of  the  root  is  5,  since  157000  is  between  503  and 
603.  Subtract  503  from  the  given  number,  and  the  remainder  will 
be  32464. 

Now  this  remainder  must  consist  of  3  x  502  x  units'  digit  +  3 
X  50  x  sq.  of  units'  digit  +  cube  of  units'  digit,  and  the  first  of 
these  three  terms  is  the  largest ;  therefore  if  we  use  3  x  502  as  a 
trial  divisor,  we  obtain  a  quotient,  namely  4,  which  is  equal  to,  or 
greater  than,  the  unknown  (units')  digit.     If  now  we  add  to  the 

.      157'464(50  +  4 
50*  =  125  000 
3  x  502         =  75J0  j  32  464* 

3  x  50  x  4*^=    600 

4*=      16 


3  x  502  +  3  x  50  x  4  +  42  32  464 


trial  divisor  the  last  two  of  the  above  three  terms  (omitting  the 
units'  digit  once  as  a  factor),  we  shall  have  as  a  true  divisor 
3  x  502  +  3  x  50  x  4  +  42  =  gll6.  Multiplying  this  by  the  units' 
digit  and  subtracting  the  product  from  32464,  we  have  no 
remainder. 

Ex.  2.  Find  the  cube  root  of  13312063.  „~ 

..     13'312'053(2H|+  30  +  7 


,  2003  = 

8^)00"  000 

3  x  2002  =  120000 

3  x  200  x*30  =#18000 

302  _   900 

5  312  e#e 

.  138900  - 

4  167  000' 

3  x  2302  £.  158700. 
3  x  230  x  7  =  4830  ' 
72=    49 

1  145  053 

163579 

1  145  053 

Here  there  are  three  periods,  and  therefore  three  figures  in  the  root ; 
and,  since  13000000  lies  between  2003  and  3003,  the  first  figure  of 
the  root  is  2.  Subtract  2003,  and  the  remainder  is  5312053.  Now 
take  3  x  2002,  that  is  120000,  as  a  'trial  divisor'  ;  and  5312053  h- 


300  CUBE   ROOT.  [Chap.  XV. 

120000  will  give  40  for  quotient.  It  will,  however,  be  found  on 
trial  that  40  is  too  great,  for 

(3  x  2002  +  3  x  200  x  40  +  402)  x  40 

is  greater  than  the  remainder  5312053  ;  we  therefore  try  30.     Take 

3  x  2002  +  3  x  200  x  30  +  302, 

and  multiply  this  sum  by  30  and  subtract  the  product  from  5312053 ; 
we  shall  then  have  subtracted  altogether  (200  +  30) 3  from  the 
given  number,  and  the  remainder  will  be  found  to  be  1145053. 

To  find  the  last  figure  of  the  root  use  3  x  2302,  that  is  158700,  as 
a  'trial  divisor,'  and  1145033  +  158700  gives  7  for  quotient.  Take 
3  x  2302  +  3  x  330  x  7  -|-  72,  aid  multiply  this  sum  by  7,  and  sub- 
tract the  product  from  1145053.  There  is  now  no  remainder ;  and, 
from  Art.  219,  we  have  now  sub#acted  altogether  (230  +  7)3 ; 
hence  the  given  number  =  2373,  so  that  237  is  the  required  cube 
root. 

Ex.  3.    Find  the  cube  root  of  252.435968. 

The  pointing  must  be  begun  from  the  units'  figure,  and  carried 
forwards  for  the  integral  part  and  backwards  for  the  decimal  part. 

252'.435'968'(6  +  .3  +  .02 
68  =  216. 
I?x62  =  108  36.435968 


3  x  6  x  .3  =    .5.4 
(.3>,2= .09 


113.49 


34.04^ 


3  x  (6.3)2=  119.07 
3  x  6.3  x  .02  =       .378 
(.02)2  =        .0004 


119.4484 


2.388968 


2.388968 


The  process  can  be  somewhat  shortened,  as  in  Square 
Root ;  it  is,  however,  very  rarely  necessary  to  find  a  cube 
root,  and  it  is  therefore  undesirable  to  attempt  to  shorten 
the  ahove  process. 


Art.  283.] 


EXAMPLES. 


301 


EXAMPLES   CXV. 

Find  the  cube  root  of  each  of  the  following  numbers : 


1.  1331. 

2.  3375. 

3.  4913. 

4.  12167. 

5.  29791. 

6.  68921. 


7.  79507000. 

8.  148877000. 

9.  8869743. 

10.  733870808. 

11.  2352637. 

12.  16974393. 


13.  2.197. 

14.  .004913. 

15.  .238328. 

16.  125525.735343. 


17.    2io. 


19.   12568-g-V    ♦ 


18.    39^. 
20.    240tfftf 
Find  to  three  significant  figures : 
21.      ^/10.     22.    ^1.5.     23.    .s/3.75.     24.    ^/.0675. 

25.  Find  the  side  of  a  cube  wkich  has  the  same  vol- 
ume as  a  beam  40  ft.  6  in.  long,  1  ft.  4  in.  wide,  and  f  in. 
thick. 

26.  Find  the  length  of  one  edge  of  a  cube  whose  vol- 
ume is  2  cu.  yd.  14  cu.  ft.  145  cu.  in. 

27.  Find  the  area  of  each  face  of  a  cube  whose  volume 
is  5  cu.  yd.  2  cu.  ft.  1592  cu.  in. 

28.  Find  approximately  the  length  of  one  edge  of  a 
cubical  vessel  which  contains  a  gallon. 

29.  Find  approximately  the  side  of  a  cube  of  iron 
which  weighs  a  t,  assuming  that  a  cu.  ft.  of  iron  weighs 
486  lb. 

30.  Find,  to  the  nearest  mm,  the  length  of  a  cube  of 
gold  which  weighs  as  much  as  a  cum  of  water,  the  S.G. 
of  gold  being  19.5. 


302  REVIEW.  [Chap.  XV. 

MISCELLANEOUS  EXAMPLES  FOR  GENERAL  REVIEW. 

1.  Express  in  words  5006017,  and  in  figures  thirteen 
million  twenty-five  thousand  eleven. 

2.  Find  the  least  multiple  of  3157  which  is  greater 
than  a  million. 

3.  How  many  articles  each  worth  $14.45  should  be 
given  in  exchange  for  60  articles  each  worth  $49.13  ? 

4.  Reduce  5 1.  7  cwt.  30  lb.  11  oz.  to  oz. 

5.  Find  the  G.C.M.  also  the  L.C.M.  of  3432  and  3575. 

6.  Find  the  sum  of  -J-,  f,  £,  j-i,  and  ^|. 

7.  Divide  43^  by  28T2^-,  and  express  the  result  as  a 
fraction  of  12. 

8.  Divide  .221312  by  5.32. 

9.  Add  .375  of  13s.  4c7.  and  .07  of  £2.  10s.,  and  sub- 
tract the  result  from  £.45. 

10.    Find  the  rent  of  134  A.  145  sq.  rd.  at  $19.50  per 
acre. 


11.  Multiply  905741  by  518963,  and  express  the  result 
in  words. 

12.  A  certain  number  was  divided  by  77  by  short  divis- 
ions ;  the  quotient  was  137,  the  first  remainder  was  9  and 
the  second  remainder  was  6 ;  what  was  the  dividend  ? 

13.  Reduce  15  m.  95  rd.  3  yd.  to  in. 

14.  A  grocer  mixed  48  lb.  of  tea  which  cost  him  64  ct. 
a  lb.  with  a  certain  quantity  which  cost  60  ct.  a  lb.  He 
then  sold  the  whole  for  $76.92,  and  gained  $7.20  by  the 
transaction.     How  much  tea  did  he  sell  ? 

15.  Express  756,  1155,  and  1176  as  the  products  of 
prime  factors. 


Art.  283.]  EXAMPLES.  308 

16.  Simplify  4i  +  i-3| +  5 J- 6H- 

17.  Simplify  |  of  £  of  4&  -  1|  of  ^. 

18.  Simplify  2.9015  x  .01702  x  .005803. 

19.  Find  f  of  £2.  lis.  lid.  -  .115625  of  £1  +  .75  d. 

20.  A  farm  of  500Ha  91a  is  rented  at  $3.25  per  Ha.; 
what  is  the  whole  rent  ? 


21.  Find  the  difference  between  seventy-six  million 
eight,  and  four  hundred  ninety-nine  thousand  four  hun- 
dred forty ;  and  divide  the  result  by  ninety-nine. 

22.  What  is  the  greatest  number  which  will  divide 
2000  with  remainder  11,  and  will  divide  2708  with  re- 
mainder 17  ? 

23.  Multiply  190  rd.  9  in.  by  144. 

24.  Taking  the  average  length  of  a  lunar  month  from 
full  moon  to  full  moon  to  be  29.5306  da.  and  the  length 
of  a  yr.  to  be  365.2422  da.,  show  that  4131  lunar  months 
are  very  nearly  equal  to  334  yr. 

25.  What  is  the  least  number  of  gr.  which  is  an  ex- 
act number  both  of  lb.  Troy  and  of  lb.  Avoir.  ?  If  the 
number  of  lb.  Troy  in  a  certain  weight  exceed  the  number 
of  lb.  Avoir,  by  496,  what  is  that  weight  in  gr.? 

26.  Simp]ify3-iA|AM_^^8|f. 

27.  Find  the  value  of  a  property  if  the  owner  of  f-  of 
it  can  sell  f  of  his  share  for  $492. 

28.  Divide  .00625  by  2500,  and  6.25  by  .0025. 

29.  Express  20  lb.  8  oz.  9  dwt.  6  gr.  as  a  decimal  of 
2541b.  10  oz. 


304  REVIEW.  [Chap.  XV. 

30.  Find  the  difference  between  the  value  of  13  cwt. 
74  lb.  of  sugar  at  $5  per  cwt.,  and  that  of  52  lb.  12  oz.  of 
tobacco  at  $120  per  cwt. 


31.  Write  MDCCCXCIX  in  Arabic  figures,  and  express 
1489  by  means  of  Roman  numerals. 

32.  A  man  takes  100  steps  a  minute,  and  the  average 
length  of  his  step  is  30  in. ;  how  far  will  he  walk  in 
4hr.? 

33.  How  much  coal  is  required  to  supply  12  fires  for 
27  weeks,  each  fire  consuming  1  cwt.  42  lb.  of  coal  weekly  ? 

34.  Find  the  greatest  number  by  which  when  4344  and 
5943  are  divided  the  remainders  will  be  31  and  41  respec- 
tively. 

35.  Simplify  267f  of  if  x  (f  -  f  -  TV). 

41-34;         41of4| 

36.  Slmphfy  nL-i+I^F-1Xf 

37.  If  if  of  4  of  29£  of  a  certain  sum  is  $1692.60, 
what  is  the  sum  ? 

38.  Reduce  .63,  .48324,  and  .01654  to  common  fractions 
in  their  lowest  terms. 

39.  What  decimal  of  $2.25  is  $5  ?  Find  the  value 
of  .78125  of  $4  -  .0625  of  $1.20  -  2.75  of  $.04. 

40.  What  is  the  cost  of  a  silver  cup  weighing  2  lb.  5  oz. 
17  dwt.  12  gr.  at  $1.85  per  oz.  ? 


41 .  $603.42  is  to  be  divided  equally  among  226  people ; 
how  much  will  each  receive  ? 

42.  The  heights  of  5  boys  are  respectively  5  ft.  41  in., 
5  ft.  2  in.,  5  ft.  11  in.,  4  ft.  10  in.,  and  4  ft.  8£  in.;  what 
is  the  average  height  ? 


Art.  283.]  EXAMPLES.  305 

43.  Reduce  726314  in.  to  mi.,  rd.,  etc. 

44.  A  circular  running  path  is  902  yards  round.  Two 
men  start  back  to  back  to  run  round,  and  one  runs  at  the 
rate  of  10  miles  and  the  other  at  the  rate  of  10^  miles  an 
hour.    When  and  where  will  they  meet  for  the  first  time  ? 

45.  Find  the  greatest  length  of  which  both  42  yd.  9  in. 
and  55  yd.  9  in.  are  multiples. 

46.  Find  the  least  fraction  which  added  to  the  sum  of 
-i-J,  ^,  and  |~§  will  make  the  result  an  integer. 

47.  What  fraction  of  $27  is  T9T  of  $1.21  ? 

48.  Simplify  ^f  x  ^55 .  and  divide  .72  by  il7936, 

.018         .64 
expressing  the  result  as  a  recurring  decimal. 

49.  Find  the  value  of  .05  of  ioi  of  £74.  18s.  6cl 

50.  A  person  buys  5  cwt.  46  lb.  of  sugar  at  $3.87^  per 
cwt.,  and  sells  it  at  4  ct.  per  lb;  what  is  the  gain  ? 


51.  Find  the  sum  of  all  the  numbers  between  100  and 
200  which  are  divisible  by  13. 

52.  If  a  person's  income  be  $1700  a  year,  find  what 
he  will  save  in  4  yr.  after  spending  on  an  average  $25.50 
a  week,  taking  52  weeks  to  a  yr.  ' 

53.  Divide  69  mi.  319  rd.  2  yd.  1ft.  10  in.  by  136. 

54.  The  L.C.M.  of  two  numbers  is  11160,  the  Gr.C.M. 
is  15,  and  one  of  the  numbers  465;  what  is  the  other 
number  ? 

55.  Simplify  lof  l^of  4|-5-^-of  4of  3f 

56.  Express as  a  simple  fraction. 

3+ ? 


306  BEVIEW.  [Chap.  XV. 

57.  Subtract  |f  of  J-  of  $21  from  |  of  ^  of  $20;  and 
express  the  difference  as  a  fraction  of  an  eagle. 

58.  Divide  .37592  by  .0125,  and  3759.2  by  .000125. 

59.  Express  }  of  2.624H1  -  *£  of  1.3751  as  Kl.     Is  the 

answer  numerically  the  same  as  cubic  meters  ? 

60.  Find  the  value  of  7  A.  80  sq.  rd.  of  land  at  $200 
per  A. 

61.  How   many  mi.,  etc.,    are   there   in    a    hundred 

million  in.  ? 

62.  If  butter  be  bought  at  $27  per  cwt.  and  sold  at 
33  ct.  per  lb.,  how  much  will  be  gained  on  every  cwt.  ? 

63.  Find  the  G.C.M.  of  1035,  391,  and  598. 

64.  Simplify  4J  of  3J  -  2£  ■*-  5^  +  6T6T  -*r  3^-. 

mm     a.      „.  2  ,  1551.65  v    21 

65 .  Simplify —  and  X 


65.1         20.02 

66.  A  square  cistern  is  3m  long  inside  and  when  filled 
contains  47.25 T  of  water;  what  is  the  depth  of  the 
cistern  inside  ? 

67.  If  a  bankrupt  pays  $23  in  a  hundred,  how  much 
will  a  creditor,  to  whom  he  owes  $7866,  receive  ? 

68.  If  8  cwt.  20  lb.  cost  $20.50,  what  would  a  t.  cost 
at  the  same  rate  ? 

69.  The  distance  between  two  stations  is  234  mi.  160  rd. 
38  yd.  2  ft.  An  engine  wheel  revolves  142878  times  in 
traveling  from  one  station  to  the  other.  How  many 
in.  does  it  travel  in  one  revolution  of  the  wheel  ? 

70.  What  is  the  greatest  weight  of  which  both  2  t. 
4  cwt.  18  lb.,  and  5  t.   5  cwt.  94  lb.  are  multiples  ? 


Art.  283.]  EXAMPLES.  307 

71.  The  sum  of  the  ages  of  a  father  and  of  his  son  is 
now  88  yr.,  and  12  yr.  ago  the  father  was  three  times  as 
old  as  the  son ;  how  old  are  they  ? 

72.  A  number  is  divided  by  210  in  three  steps,  the 
factors  being  5,  6,  7  in  order;  and  the  remainders  are 
2,  3,  4  in  order ;  what  would  have  been  the  remainders 
if  the  number  had  been  divided  by  7,  6,  5  in  order  ? 

73.  Reduce  216875  in.  to  mi.,  etc.,  also  57637  sq.  yd. 
to  A.,  sq.  rd.,  and  sq.  yd. 

74.  Find,  to  within  a  thousandth  of  the  whole,  the 
square  roots  of  15,  -^,  and  .081. 

75.  Find  two  numbers,  one  of  which  is  double  the  other, 
and  whose  product  is  8192. 

76.  Find  the  following : 

17  x  19,  18  x  14,  13  x  16,  19  x  15, 
752,  952,  1052,  1152. 

77.  Find  the  least  length  which  is  a  multiple  of  1  ft. 
6  in.,  4  ft.  6  in.,  7  ft.  6  in.,  and  15  ft.  9  in. 

78.  What  is  the  acreage  of  a  rectangular  field  whose 
sides  are  respectively  201  yd.  2  ft.,  and  60  yd.  ? 

79.  A  rectangular  field  contains  2  A.  134  sq.  rd.,  and 
its  length  is  6.25  ch. ;  what  is  its  breadth  ? 

80.  What  is  the  least  length  of  carpet  27  in.  wide 
that  would  be  required  to  cover  the  floor  of  a  room  24  ft. 
loner  and  21  ft.  wide  ? 


81.  Simplify   4f  -  (^  x  ^  X  10*)  and  4f  -  ^  x  T\ 
xlOi 

82.  Simplify  7  x  16  -  }  of  4*  -  1\  x  17  -  \  4  (18  -  6) 
+  (26-3)|-7. 


308  REVIEW.  [Chap.  XV. 

83.  Find  by  factors  ^1936,  V2601,  Vdrffir- 

84.  How  much  will  it  cost  to  paint  the  ceiling  of  a 
room  15  ft.  6  in.  long  and  12  ft.  6  in.  wide  at  16  ct.  per 
square  foot  ? 

85.  How  many  loads  (cu.  yd.)  of  gravel  would  be  re- 
quired to  cover  to  a  depth  of  2  in.  a  path  90  yd.  long 
and  5  ft.  wide  ? 

86.  One  side  of  a  square  field  of  22 J  A.  abuts  on  a 
road.  This  side  is  divided  into  building  plots  100  ft. 
deep  and  having  a  frontage  along  the  road  of  30  ft.  each. 
The  building  plots  are  let  at  £12  each,  and  the  rest  of 
the  field  at  £5. 10s.  an  A.  What  is  the  total  rental  of 
the  property  ? 

87.  A  dealer  purchased  40  tubs  of  butter,  each  contain- 
ing 35  lb.,  at  22  ct.  per  lb.,  and  sold  35  tubs  of  the  butter 
for  as  much  as  the  whole  cost;  for  how  much  per  lb. 
must  he  sell  the  remainder  in  order  to  gain  16  %  and 

$3.22  ? 

88.  What  is  the  acreage  of  a  rectangular  field  whose 
length  is  117  rd.  and  whose  breadth  is  55  rd.  ? 

89.  The  number  of  sheep  on  a  farm  increased  for  4  yr. 
at  the  rate  of  20%  each  year,  and  there  were  originally 
625  sheep ;  how  many  were  there  at  the  end  of  the  4  yr.  ? 

90.  A  man  makes  a  profit  of  20%  by  selling  an  article 
for  24  ct. ;  how  much  %  would  he  make  by  selling  it  for 
25  ct.  ? 

91.  From  a  vessel  containing  32.3 J  of  kerosene  1722.5 cl 
were  drawn ;  how  many  dl  remained  ? 

92.  Find  the  least  number  which  when  divided  by  17 
leaves  a  remainder  12,  and  when  divided  by  29  leaves  a 
remainder  24. 


Art.  283.]  EXAMPLES.  309 

93.    Reduce  to  their  simplest  forms  : 

©  T3T(i  +  f  +  iV-i)-ii°f^r 

(")    <*  +  *+*  +  €>+<*+*+*+» 

94.  The  age  of  a  father  is  three  times  the  sum  of  the 
ages  of  his  three  sons,  and  two  years  ago  the  father's  age 
exceeded  the  sum  of  the  ages  of  the  three  sons  by  36 
years ;  how  old  is  the  father  ? 

95.  A  body  weighs  60 g  in  air  and  42  g  in  water;  what 
is  its  S.G.  ? 

96.  Find  the  interest  on  a  30  da.  Mass.  note  for 
$7895.56. 

97.  A  man  bought  13$  bu.  of  corn  for  $7.77,  and 
sold  the  same  at  20%  profit;  what  was  the  selling  price 
per  bu.  ? 

98.  A  man  paid  $45.10,  including  a  duty  of  10%,  for 
a  watch ;  how  much  was  the  duty  ? 

99.  The  distance  between  two  places  on  a  map  is 
156 mm;  what  is  the  distance  in  Km  if  the  scale  of  the 
map  is  1  to  80000  ? 

100.   Find  the  number  of  Km  in  one  mi. 


101.  Find   the  prime  factors  of   the  L.C.M.  of  391 
and  493. 

102.  If  15%  be  lost  by  selling  an  estate  for  $3400,  for 
what  must  it  be  sold  to  gain  20%  ? 

103.  Find  V-6  *°  the  nearest  thousandth. 


310  REVIEW.  [Chap.  XV. 

104.  A  beam  36  ft.  long,  and  whose  section  is  a  square, 
contains  182-J-  cu.  ft.  of  timber ;  what  is  its  width  ? 

105.  Find  the  length  of  the  side  of  a  square   field 
which  contains  10  A. 

106.  Divide  570326  by  63  by  'short'  divisions,  ex- 
plaining clearly  the  formation  of  the  remainder. 

107.  Reduce  to  its  simplest  form 

ft  +  l)of  (f+iHfof  (i  +  1vH||^fi 

108.  Multiply   36.2   by   .057,   and    divide   5752.8  by 
.00376,  and  .0025  by  3.1. 

109.  Find  the  value  of  a  bar  of  gold  weighing   5  lb. 
10  oz.  17  dwt.  22  gr.  at  $20  per  oz. 

110.  How  many  gallons  will  a  cistern  6  ft.  by  4  ft.  by 
3ft.  hold? 


111.  The  total  number  of  votes  given  for  two  candi- 
dates at  an  election  was  127345,  and  the  successful 
candidate  had  a  majority  over  the  other  of  17377;  how 
many  votes  did  each  get  ? 

112.  Divide  $875  between  three  persons  so  that  the 
first  may  have  $50  more  than  the  second,  and  the  second 
$75  less  than  the  third. 

113.  A  certain  number  less  than  1000,  when  divided 
by  56  or  by  72  leaves  13  as  remainder;  what  is  the 
number  ? 

114.  A  grocer  mixes  91b.  of  coffee  at  54  ct.  alb.  with 
6  lb.  of  chicory  at  15  ct.  a  lb ;  at  what  price  per  lb.  must 
he  sell  the  mixture  in  order  to  get  a  profit  of  25%? 


Art.  283.]  EXAMPLES.  311 

115.  The  breadth  of  a  room  is  twice  its  height  and  the 
length  is  thrice  its  height ;  and  it  cost  $115.20  to  paint 
the  walls  at  $.08  per  sq.  ft. ;  what  is  the  height  ? 

116.  How  many  turfs  each  3  ft.  by  1  ft.  would  be  re- 
quired to  turf  a  lawn  96  ft.  by  75  ft.,  and  how  much 
would  they  cost  at  $1.75  a  hundred  ? 

117.  Find  the  weight  of  a  rectangular  solid  piece  of 
iron  17 cm  by  5cm  by  3cm,  the  S.G-.  of  iron  being  7.8. 
Answer  in  Kg. 

118.  Find  the  interest  on  $672.87  for  2  yr.  7  mo. 
at  4%. 

119.  In  a  room  22  ft.  by  18  ft.  there  is  a  Turkey 
carpet  with  a  border  2  ft.  wide  all  round  it.  The 
carpet  cost  20  ga. ;   how  much  was  that  a  sq.  yd. 

120.  Find  the  length  of  a  square  field  whose  area  is 
4  A.   89  sq.  rd. 

121.  A  wire  .2346  yd.  long  is  cut  up  into  pieces  each 
.007  yd.  long ;  how  many  pieces  will  there  be,  and  what 
length  will  be  left  over  ? 

122.  A  room  is  21  ft.  long,  17  ft.  wide,  and  12  ft.  high ; 
how  many  pieces  of  paper  21  in.  wide  and  12  yd.  long 
must  be  bought  to  paper  the  room  supposing  150  sq.  ft. 
of  the  walls  are  left  uncovered  ? 

123.  A  class  contains  19  boys;  and  in  an  examination 
6  boys  got  56%  of  the  full  marks  each,  one  got  90%,  and 
the  rest  got  39%  each,  except  one  boy  who  got  no  marks 
at  all ;  what  was  the  average  %  got  by  the  boys  in  the 
class  ? 

124.  A  rectangular  block  of  timber  is  5  ft.  long  and 
contains  3  cu.  ft.  If  its  section  be  a  square,  find  its 
thickness  to  the  nearest  tenth  of  an  in. 


312  REVIEW.  [Chap.  XV. 

125.  A  square  field  is  bordered  by  a  path  one  yd. 
wide,  the  field  and  path  together  occupying  two  and 
one  half  A. ;  find  the  cost  of  covering  the  path  with 
gravel  at  36  ct.  per  sq.  yd. 

126.  A  flask  holding  25ccm  of  water,  holds  20.25 «  of 
alcohol ;  find  the  S.G.  of  the  alcohol. 

'  127.    Some  goods  cost  $25 ;  how  much  is  lost  by  selling 
them  at  20%  below  cost  ? 

128.  One  lb.  Troy  is  what  %  of  one  lb.  Avoir.  ? 

129.  What  is  the  proceeds  of  a  N.  Y.  note  for  $2040 
drawn  Jan.  31,  '95,  at  3  mo.  and  discounted  on  Feb.  25th 
at  5%  ? 

130.  What  sum  is  invested  if  the  investment  yields 
$585  per  annum  at  4^%  ? 


131.  Eeduce  563147  in.  to  mi.,  etc. 

132.  Find  the  prime  factors  of  58212.  What  is  the 
greatest  square  number  of  which  58212  is  a  multiple  ? 

133.  Simplify    2W  +  *(*  +  *)-J  (*  +  *) 

Joff-foff 

134.  Find  the  acreage  of  a  rectangular  field  whose 
length  is  25  ch.  80  li.  and  whose  breadth  is  8  ch.  75  li. ; 
find  also  the  rent  at  $12  an  A. 

135.  A  certain  piece  of  work  can  be  done  by  8  men  or 
16  boys  in  10  da.  In  how  many  da.  can  the  work  be 
done  by  8  men  and  16  boys  ? 

136.  An  object  weighs  10  g  in  air  and  4g  in  water ;  find 
its  S.G. 


Art.  283.]  EXAMPLES.  313 

137.  A  man,  after  deducting  $4000  from  his  income, 
pays  $170  income  tax  on  the  remainder.  If  the  $4000 
had  not  been  deducted,  the  tax  would  have  been  $250. 
Find  the  rate  of  taxation  and  the  income. 

138.  A  demand  note  with  interest  was  paid  4  yr.  after 
date.  The  interest  at  ±\  %  was  $365.04;  find  the 
principal. 

139.  A  demand  note  bearing  interest  was  paid  4  yr. 
after  date.  The  amount  at  5%  was  $2433.60;  find  the 
principal. 

140.  A  train  110  yd.  long  was  observed  to  pass  a 
certain  point  in  10  sec. ;  how  many  mi.  an  hr.  was  it 
then  going  ? 

141.  Determine  the  number  which  when  divided  by 
231  by  the  method  of  '  short '  divisions,  gives  a  quotient 
583,  and  2,  6,  and  10  as  successive  remainders. 

142.  Find  the  G.C.M.  of  464321  and  683111,  and  hence 
find  all  the  common  measures  of  those  numbers. 

143.  Find  the  weight  in  Kg  of  the  air  in  a  room  60  ft. 
long,  36  ft.  wide,  and  21  ft.  high,  assuming  that  one 
cu.  yd.  =  .765cura,  and  that  air  weighs  1.29 g  per  liter. 

144.  The  wages  of  A  and  B  together  for  45  da. 
amount  to  the  same  sum  as  the  wages  of  A  alone  for  72 
da. ;  for  how  many  da.  will  this  sum  pay  the  wages  of 
B  alone  ? 

145.  A  room  is  20  ft.  7  in.  long,  15  ft.  5  in.  wide,  and 

11  ft.  high.     Find  the  number  of  pieces  of  paper,  each 

12  yd.  long  and  21  in.  wide,  which  would  have  to  be 
bought  to  paper  the  walls,  supposing  that  windows, 
etc.,  which  are  not  papered,  make  up  one-sixth  of  the 
whole  surface  of  the  walls. 


314  EEVIEW.  [Chap.  XV. 

146.  Ten  loads  (cubic  yards)  of  gravel  are  spread  uni- 
formly over  a  path  180  ft.  long  and  4  ft.  wide ;  what  is 
the  depth  of  the  gravel  ? 

147.  A  merchant  borrowed  $2000  from  a  Philadelphia 
bank  for  30  da.  at  5%  ;  find  the  proceeds  of  the  note. 

148.  An  Ohio  farmer  sold  some  sheep  for  $475,  and  took 
in  payment  a  3  mo.  interest-bearing  note  dated  Jan.  6,  '93, 
rate  h\%.  On  Mch.  1  st  the  farmer  had  the  note  discounted 
at  5%  ;  how  much  cash  did  he  receive  from  the  bank  ? 

149.  One  pound  of  silver  is  weighed  in  water;  how 
many  pwt.  does  it  lose,  the  S.G.  being  10.5  ? 

150.  Find  the  weight  in  dg  of  a  cylindrical  stick  of 
silver  10 cm  long  and  lcm  in  diameter,  the  S.G.  being  10.5. 


151.  Find  the  least  length  which  is  a  multiple  of  5  yd. 
1  ft.  3  in.,  and  also  of  7  yd.  2  ft.  9  in. 

152.  Simplify (4i-2f of ^  + 2i)-K4i-2|)off-f2^. 

153.  Find  V783  to  tne  nearest  tenth. 

154.  (i)  Multiply  17  +  19  +  16  by  18. 
(ii)  Find  mentally  (30  +  4)2. 

(iii)  Find  mentally  852. 

155.  Find 

[352  -*-  72  x  42  -  J  (150  x  §  -s-  25)  +  1260  +  35|]  11. 

156.  (i)  A  ratio  is  47 ;  find  the  second  term  when  the 
first  term  is  235. 

(ii)  A  ratio  is  -^ ;  find  the  first  term  when  the  second 
term  is  -fa. 

(iii)  Two  similar  rooms  are  respectively  8  yd.  and  9  yd. 
long ;  how  much  paper  will  be  required  to  paper  the  first 
room,  compared  with  that  which  will  be  required  for  the 
second  room  ? 


Art.  283.]  EXAMPLES.  315 

157.  Sound  travels  at  the  rate  of  1090ft.  a  second; 
how  far  off  is  a  thunder-cloud  when  the  sound  follows 
the  flash  after  5|  sec.  ?  Answer  to  the  nearest  hun- 
dredth of  a  mi. 

158.  A  father,  who  had  three  children,  left  his  second 
son  $500  more  than  he  left  the  third  son,  and  his  eldest 
son  twice  as  much  as  the  third.  They  had  $8500  between 
them ;  how  much  had  each  ? 

159.  In  a  certain  examination  every  candidate  took 
either  Latin  or  Mathematics,  also  79.4%  of  the  candi- 
dates took  Latin  and  89.6%  took  Mathematics.  If  there 
were  1500  candidates  altogether,  how  many  took  both 
Latin  and  Mathematics  ? 

160.  For  what  sum  must  goods  worth  $6370  be  insured 
at  2%  premium  so  that  in  case  of  loss  the  owner  may 
recover  the  value  both  of  the  goods  and  the  premium  ? 


161     SimDlifv     ^ofl-Sjof^ 
161-    Smphfy   ^(A-Do!^ 

162.  A  bill  of  $301.05  was  paid  with  an  equal  number 
of  eagles,  dollar  pieces,  quarters,  and  five-cent  pieces; 
how  many  coins  of  each  kind  were  there  ? 

163.  A  and  B  received  respectively  ^  and  -fa  of  a 
certain  sum  of  money,  and  C  received  the  remainder. 
A  received  $1173;   how  much  did  C  receive? 

164.  What  is  the  cost  of  a  plot  of  building-land  242 
ft.  long  and  21  ft.  wide  at  $2000  an  A.  ? 

165.  At  the  beginning  of  a  year  the  population  of  a 
town  was  16400.  The  deaths  during  the  year  were  3% 
of  the  population  at  the  beginning  of  the  year,  and  80% 
of  the  births.  What  was  the  population  at  the  end  of 
the  year,  neglecting  changes  caused  by  traveling  ? 


316  REVIEW.  [Chap.  XV. 

166.  A  liter  flask  was  half  filled  with  sand,  and  the 
weight  of  the  sand  was  1375g;  what  was  the  S.G-.  of  the 
sand? 

167.  What  is  the  cost  of  concreting  the  bottom  of  a 
circular  pond  70  ft.  in  diameter,  when  concreting  costs 
11.87  per  sq.  yd.  ? 

168.  Find  the  exact  interest  on  $700  for  30  da.,  at  6%. 

169.  A  merchant  sold  goods  for  $5650,  with  20%  and 
5%  discount,  and  10%  off  for  cash.  Cash  was  paid;  how 
much  did  the  merchant  receive  for  his  goods  ? 

170.  The  buyer  of  the  goods  in  Ex.  169  sold  the  goods 
for  $5603.38 ;  what  was  his  %  profit  ?  What  was  his 
percentage  profit? 

171.  Express  1887  by  means  of  Roman  numerals. 

172.  Express  in  t.  and  fractions  of  a  t.  the  weight  of 
lead  required  to  cover  a  flat  roof,  147  sq.  yd.  in  extent, 
with  sheet  lead  one-eighth  of  an  in.  thick,  supposing 
that  a  cu.  ft.  of  lead  weighs  820  lb. 

173.  Simplify 

174.  Find  the  value  of  a  silver  cup  weighing  2  lb.  7  oz. 
7  dwt.  12  gr.  at  $1.20  an  oz. 

175.  Find  the  cost  of  painting  the  sides  and  bottom  of 
a  cistern  3  yd.  long,  5  ft.  wide,  and  3^  ft.  deep  at  3s.  9d. 
per  sq.  yd. 

176.  Two  similar  boxes  hold  125  lb.  of  sand  and  216  lb. 
of  sand  respectively ;  the  larger  box  is  36  in.  long ;  find 
the  length  of  the  smaller  box. 

177.  Find  the  bank  discount  on  a  note  for  $1460 
payable  in  San  Francisco  30  da.  after  date. 


Art.  283.]  EXAMPLES.  317 

178.  Find  the  trade  discount  on  a  bill  of  goods  for 
$1460  with  15%  and  7%  off. 

179.  The  volume  of  a  room  is  2592  cu.  ft. ;  what  is  the 
length  of  the  room  when  the  height  is  9  ft.  and  the 
breadth  is  16  ft.  ? 

180.  For  economy  which  way  would  carpet  strips  run 
in  the  room  of  Ex.  179  ? 


181.  What  number  is  the  same  multiple  of  354  that 
86445  is  of  765  ? 

182.  Subtract  f  of  f  from  1±-  of  f ;  and  divide  the 
result  by  (f  -f>x  (|  -  f). 

183.  A  lidless  cistern  10  ft.  6  in.  long,  7  ft.  4  in.  broad, 
and  5  ft.  4  in.  high  is  to  be  painted  outside ;  find  the 
cost  at  4^  ct.  per  sq.  ft. 

184.  A  promissory  note,  written  for  30  da.  and  payable 
in  Ohio  at  4%,  amounts  to  $1505.50 ;  find  the  principal. 

185.  A  promissory  note,  written  for  45  da.  and  payable 
in  Ohio  at  4%,  amounts  to  $1960.40;  what  would  have 
been  the  amount  if  the  note  had  been  payable  in  N.Y.  ? 

186.  When  railroad  4's  can  be  bought  at  101^  (broker- 
age i),  how  many  such  bonds  can  be  bought  for  $7317  ? 

187.  A  man  buys  $5000  of  Government  4's  at  lllj 
(brokerage  \)  ;  what  %  is  he  receiving  on  his  investment  ? 

188.  A  traveler  purchases  £500  at  4.88|  (commission 
3^)  ;  how  many  dollars  does  he  pay  ? 

189.  A  train  moves  6  in.  the  1st  sec,  1  ft.  the  2d  sec, 
and  so  on  for  75  sec,  and  then  moves  37^-  ft.  per  sec.  for 
1  h. ;  how  far  does  the  train  go  in  the  1  h.  1  min.  and  15 
sec  ?  (Ans.  to  the  nearest  thousandth  of  a  mile.)  The 
answer  lacks  how  many  in.  of  the  exact  result  ? 


818  REVIEW.  [Chap.  XV. 

190.    In  a  decreasing  arithmetical  progression  a  =  12, 
d  =  £,  n  =  50 ;  find  I  and  s. 


191.  The  nearest  of  the  fixed  stars  is  roughly  twenty 
trillion  mi.  distant.  Show  that  it  would  take  light  3J 
yr.  to  traverse  this  distance  at  the  rate  of  190000  mi. 
a  sec. 

192.  Simplify 

17ixl6-3-(i  +  i)J-17i-S6-3xa  +  i)J. 

193.  Express  Ida.  4hr.  31  min.  h2\  sec.  as  a  decimal 
of  3  da.  4  hr.  5  min. 

194.  Find  the  value  of  11  oz.  13  dwt.  8  gr.  of  gold  at 
$1.02  per  dwt. 

.  195.  What  will  it  cost  to  carpet  a  room  18  ft.  long  and 
15  ft.  wide,  the  carpet  being  27  in.  wide  and  costing  $1.05 
a  yd.? 

196.  A  ship  is  worth  $45000.  For  what  sum  must  it 
be  insured  at  $5  per  $100  in  order  that  the  owner  in  case 
of  loss  may  receive  the  value  of  the  ship  and  the  amount 
of  premium  paid  ? 

197.  At  what  rate  %,  simple  int.,  will  $7600  amount 
to  $7676  in  3  mo.  ?      (No  grace.) 

198.  What  is  the  price  of  a  4%  stock,  if  a  man  who 
invests  $4301  gets  an  income  of  $136  a  year  on  his  in- 
vestment ?     (Brokerage  ■£-.) 

199.  A  man  bought  $100  bonds  at  89  and  sold  them  at 
95  (brokerage  \  on  each  transaction)  and  made  a  profit  of 

>.25 ;  how  many  bonds  did  he  buy  ? 

200.  Find  the  mean  proportional  between 
(i)  4  and  36  ; 

(ii)  .25  and  ll2; 
(iii)  .64  and  1.44. 


Art.  283.]  EXAMPLES.  319 

Find  a  third  proportional  to 
(iv)  2.5  and  4.5; 
(v)  .7  and  7 ; 
(vi)  152  and  52. 


201 .  A  train  is  traveling  at  the  rate  of  35  mi.  an  hr. ; 
how  many  ft.  does  it  go  in  a  sec.  ? 

202.  Simplify  2.42  -  .0025  x  .02  --  .055. 

203.  Find  the  rent  of  375.4875  A.  at  $12.80  an  A. 

204.  How  many  loads  (en  m)  of  gravel  will  be  required 
to  cover  a  court-yard  20 m  by  15 m  to  a  depth  of  5cm, 
and  how  much  will  the  gravel  cost  at  84  ct.  a  load  ? 

205. 


fjf.75™.  Boolon,  Mom,.,  fan.  6,  18<?5. 

Svtt&eM,  cLum,  att&v  ciat&  <J  promise  to  pay  to 

the  order  of WLy&elfi 

^^.3'cyuA,  hwrucLn&cL  &&v-&nty-fCv-&^^  ^Dollars 

at tk&  &I06-&  cAatuyyual  Bank 

Value  received. 

No.  205.      Due 3vtu&  Lw&bcyvnA. 


When  was  this  note  due  ?     What  were  the  proceeds  ? 

206.  How  many  gallons  will  fall  on  a  sq.  mi.  in  a 
rainfall  of  -^  of  an  in.,  and  how  many  t.  will  the  water 
weigh  ?     (1  gal.  of  water  weighs  8.33  lb.) 


320  REVIEW.  [Chap.  XV. 

207.  A  man  has  an  income  of  £525.  5s.  from  2 J  per 
cent  consols.  He  sells  ont  at  96^,  and  buys  4  per  cent 
Kussian  bonds  of  £100  at95-J-.  What  will  be  the  change 
in  his  income  ?     (The  prices  include  brokerage.) 

208.  Find  the  exact  interest  on   $750    for  36  da.    at 

209.  Two  rooms  of  the  same  height  are  respectively 
15  ft.  and  20  ft.  square ;  what  is  the  ratio  between  the 
numbers  of  rolls  of  paper  required  for  the  walls  of  the 
rooms  ?     For  the  ceilings  of  the  rooms  ? 

210.  The  first  of  two  similar  rooms  requires  94 qm  of 
plastering  for  its  walls  and  ceiling,  and  is  6 m  long ;  how 
many  qm  of  plastering  are  required  for  the  second  room, 
which  is  7m  long?  How  many  cum  of  mortar  are  re- 
quired for  the  first  room  if  the  thickness  of  the  plastering 
be  1 cm  ? 

211.  Express   4  min.   12  sec.   as  a  decimal  of  a  week. 

212.  A  man  sold  25  articles  for  the  same  price  as  he 
paid  for  35 ;  what  was  his  profit  %  ? 

213.  The  number  of  oz.  Avoir,  in  a  certain  weight 
exceeds  the  number  of  oz.  Troy  by  17 ;  what  is  the  weight 
Avoir.  ? 

214.  If  a  number  when  divided  by  391  leaves  a  remain- 
der of  300,  what  will  be  the  remainder  when  the  number 
is  divided  by  17  ? 

215.  A  grocer  bought  tea  at  32  ct.  per  lb.  and  sold  so 
as  to  gain  25  %  ;  the  duty  on  tea  was  reduced,  and  he  then 
bought  and  sold  at  4ct.  per  lb.  less  than  before;  what 
was  his  gain  %? 

216.  A  man  invested  $38400  in  2f%  bonds  at  95£; 
how  much  stock  at  109  J-  could  he  have  bought  with  his 
first  semi-annual  interest  ? 


Art.  283.]  EXAMPLES.  321 

217.  A  man  embarks  his  whole  property  in  four  suc- 
cessive ventures.  In  the  first  he  gained  60%,  and  in 
each  of  the  others  he  lost  20% ;  what  was  his  total 
loss  %  ? 

218.  A  man  spent  one-third  of  his  income  on  lodgings, 
one-fourth  the  remainder  on  food,  one-fifth  what  was  left 
on  clothes,  one-sixth  of  the  remainder  on  books,  and  then 
had  $1200  left;  what  was  his  income  ? 

219.  If  a  railroad  stock  pays  a  7%  annual  dividend, 
at  what  price  must  the  stock  be  bought  so  as  to  yield  4% 
on  the  investment  ?     (Brokerage  as  usual.) 

220.  A  rectangular  field,  whose  area  is  1A.  65  sq.  rd., 
is  137  yd.  1  ft.  6  in.  long ;  what  is  its  breadth  ? 


221.  A  cistern  9  ft.  long,  8  ft.  broad,  and  6  ft.  deep  is 
supplied  with  water  by  a  pump  which  will  send  in  27 
gal.  a  min. ;  how  long  will  it  take  to  fill  the  cistern  ? 
(Answer  to  the  nearest  sec.) 

222.  A  cistern  3m  long,  2.5m  broad,  and  2m  deep  is 
supplied  by  a  pipe  through  which  run  150 '  of  water  per 
minute ;  how  many  minutes  will  be  required  to  fill  the 
cistern  ? 

223.  In  the  centre  of  a  room  23  ft.  square  there  is  a 
carpet  18  ft.  square  and  the  rest  of  the  floor  is  covered 
with  oil-cloth  which  is  extended  6  in.  under  the  carpet. 
The  carpet  cost  $2.25  a  yd.  and  the  oil-cloth  cost  90  ct.  a 
sq.  yd. ;  what  was  the  whole  cost  ? 

224.  A  man  leaves  by  will  $3600  to  his  wife,  and  the 
remainder  of  his  property  to  be  equally  divided  between 
his  four  children ;  and  it  was  found  that  the  share  of  each 
child  was  one-seventh  of  the  whole  property ;  how  much 
did  the  man  leave  ? 


324  REVIEW.  [Chap.  XV. 

244.  In  the  centre  of  a  square  court  is  a  square  of 
grass  covering  T9g  of  the  whole  area  of  the  court,  and  the 
side  of  the  square  of  grass  is  60  feet;  find  the  cost  of 
graveling  the  remainder  of  the  court  to  a  depth  of  3 
in.,  the  gravel  and  labor  costing  $  1.08  a  cu.  yd. 

245.  A  man  buys  eggs  at  30  ct.  per  dozen,  and  sells 
them  at  $2.80  per  hundred;  what  is  his  gain  %? 

246.  A  grocer  pays  24  ct.,  30  ct.,  and  40  ct.  per  lb. 
respectively  for  three  different  kinds  of  tea.  If  he  mixes 
weights  of  these  teas  proportional  to  the  numbers,  6,  4, 
and  3,  respectively,  and  sells  the  mixture  at  36  ct.  per 
lb.,  what  profit  does  he  make  %? 

247.  A  piece  of  work  can  be  done  in  48  da.  by  15 
men,  but  after  9  da.  two  of  the  men  go  away;  in  how 
many  more  da.  will  the  men  who  remain  finish  the 
work? 

248.  By  selling  goods  for  $45.60  a  man  lost  5%  ; 
what  would  he  have  gained  if  he  had  sold  for  $57  ? 

249.  A  dealer  bought  a  certain  number  of  articles  at 
the  rate  of  40  in  a  lb.,  and  twice  the  same  number  at 
the  rate  of  50  in  a  lb.  He  sold  the  whole  at  the  rate  of 
36  in  a  lb. ;  how  much  %  did  he  gain  ? 

250.  At  $1.12|  per  sq.  yd.,  it  cost  $506.25  to  carpet  a 
room  whose  length  is  double  its  breadth,  and  whose 
height  is  §  its  breadth ;  how  high  is  the  room  ? 


251.  Simplify 

65J/3H     40\  /4j  +  6f\  | 

569K25+4iJ      n^7A-3|;-32^|- 

252.  A  clock  is  set  right  at  noon,  but  when  it  strikes 
12  that  night  it  is  80  sec.  fast ;  find  how  many  minutes 
it  will  gain  in  a  week. 


Art.  283.]  EXAMPLES.  325 

253.  In  a  race  of  100  yards  A  can  beat  B  by  5  yards, 
B  can  beat  C  by  5  yards,  and  C  can  beat  D  by  5  yards ; 
how  should  they  be  handicapped  for  a  100  yards'  race, 
putting  A  at  scratch  and  giving  him  the  advantage  of 
any  odd  fraction  of  a  foot  ? 

254.  What  is  the  least  number  which  when  divided  by 
15  leaves  a  remainder  9,  when  divided  by  35  leaves  a 
remainder  29,  and  when  divided  by  42  leaves  a  remain- 
der 36  ? 

255.  The  commercial  discount  and  interest  on  a  cer- 
tain sum  for  the  same  time  and  rate  are  $254.10  and 
$252  respectively ;  find  the  sum. 

256.  A  man  invests  $9875  partly  in  a  3%  stock  at 
104 J  and  partly  in  a  5%  stock  at  152 f,  and  he  obtained 
2-f|%  on  his  outlay;  how  much  did  he  invest  in  each 
stock  ? 

257.  Three  persons,  A,  B,  and  C,  working  together  com- 
plete a  piece  of  work  which  it  would  have  taken  them 
respectively  9,  10,  and  12  da.  to  complete  if  working 
separately.  They  receive  in  payment  $25.44,  which  they 
are  to  divide  in  proportion  to  the  quantity  of  work  done 
by  each ;  find  their  shares. 

258.  Which  term  of  the  series,  9,  12, 15,  etc.,  is  636  ? 

259.  Find  (90  +  2)2 ;  (60  +  7)2 ;  (40  +  8)2. 

260.  A  man  had  4%  Railroad  Preferred  Stock  which 
brought  him  $664  a  yr.  He  sold  out  at  119-J-,  and  invested 
in  Common  Stock  at  145^.  The  Common  Stock  paid  6% 
dividends ;  what  was  his  gain  in  income  per  yr.  ? 


261.  A  square  field  contains  22  A.  80  sq.  rd. ;  how 
long  will  it  take  a  boy  to  run  round  the  boundary  of  the 
field  at  the  rate  of  12  mi.  an  hr.  ? 


326  REVIEW.  [Chap.  XV. 

262.  A  man  bought  a  certain  number  of  eggs  at  the 
rate  of  one  for  a  ct.,  three  times  the  number  at  the  rate 
of  three  for  two  ct.,  six  times  the  number  at  11  ct.  per 
dozen,  and  ten  times  the  number  at  the  rate  of  16  ct.  per 
score,  and  sells  them  at  the  rate  of  90  ct.  per  hundred, 
gaining  by  the  transaction  $ 3.60 ;  how  many  eggs  did  he 
purchase,  and  what  did  he  gain  %? 

263.  A  grocer  has  two  sorts  of  tea,  which  cost  him 
64  ct.  and  50  ct.  per  lb.  respectively ;  in  what  ratio  must 
he  mix  them  so  that  he  may  gain  25%  by  selling  the 
mixture  at  70  ct.  per  lb.  ? 

264.  A  room  three  times  as  long  as  it  is  broad  is 
carpeted  at  $1.08  per  sq.  yd.,  and  the  walls  are  colored  at 
18  ct.  per  sq.  yd.,  the  respective  costs  being  $39.69  and 
$20.16;  find  the  dimensions  of  the  room,  making  no 
allowance  for  doors,  etc. 

265.  A  borrows  from  B  $550  at  6%  ;  six  da.  after- 
wards B  borrows  from  C  a  certain  sum  at  6%  ;  A  pays  his 
debt  in  36  da. ;  B  pays  his  debt  in  30  da.  The  interest 
being  the  same  in  each  case,  what  was  B's  debt  ? 

266.  A  man  who  held  $24600  of  3±-'s  sold  out  at  93 
and  purchased  as  many  4's  as  possible  at  130  J.  He  sold 
the  4's  at  139§-  and  re-invested  in  3^'s  at  94.  What  was 
the  change  in  his  annual  income,  and  how  much  money 
was  not  re-invested  ? 

267.  Find  the  sum  of  51  consecutive  odd  numbers,  the 
greatest  of  which  is  117. 


Art.  283.]  EXAMPLES.  327 


268. 


$i/-670™-.        -gvzznli&U,  Ma**.,  Jb&e,.  £<?,  18 <?/. 
&hv&&  ryvo-ntAfr  a>j<t&v  dat&  c/  promise  to  pay  to 

the  order  of @Aa&.  /?.  &C&ld 

Suva,  th<yu&a,rul>  &i/>G  huAvcUs&oL  &&v-&ntu  ~  Dollars 

at^^^~~£k&  c^varMOyi  (ZLowntAf  Bank . 

Value  received,  wM,  vnt&v&bt  at  i5|%. 

JVo.  /y-2.      Due jkw^  T^IoaXaav. 


Discounted  at  5%  on  Jan.  12,  '92.     Proceeds  =  ? 


269. 


f2600^.  ftcvnycyb,  7n&.;  f<wne  6,  18  W. 

&w~&ntAf  ciaAfQy  a^t&v  dat&  J  promise  to  pay  to 

the  order  of^^^^^^^o^f  fon^a, 

otwo-  tko-ubOAul  ^iv-&  h>u,yulv&cl  c^yui  ^Dollars 
at^^XA&  /aX  cAaZio-nal  JSank^,  ffio-atvyi,  7fla&a>. 
Value  received. 
No.  /<?.      Due flf&nvy  cfwvCLfa. 


Discounted  June  19  at  4|%.    Proceeds  =  ? 

270.   Find  the  cost  in  New  York  of  a  Bill  of  Exchange 
on  Paris  for  4130/,  when  exchange  is  quoted  at  5.16^. 


328  REVIEW.  [Chap.  XV. 

271.  A  person  standing  on  a  railway  platform  noticed 
that  a  train  took  21  sec.  to  pass  completely  through  the 
station,  which  was  88  yd.  long,  and  that  it  was  9  sec.  in 
passing  him ;  how  long  was  the  train,  and  at  what  rate 
per  hr.  was  it  traveling  ? 

272.  Two  trains  start  at  the  same  time  from  A  and  B 
and  proceed  towards  each  other  at  the  rate  of  35  and  45 
mi.  per  hr.  respectively.  When  they  meet,  one  train  has 
gone  17^- mi.  further  than  the  other.  What  is  the  dis- 
tance from  A  to  B  ? 

273.  A  man  bought  a  house  and  sold  it  so  as  to  gain 
5  per  cent.  Had  he  given  10  per  cent  more  for  the  house, 
and  sold  it  for  $129.60  more  than  he  did  sell  it  for,  he 
would  have  lost  2\  per  cent.  Find  what  he  gave  for  the 
house. 

274.  An  example  in  multiplication  was  worked  cor- 
rectly, ancj.  then  all  the  figures  except  those  4 — 
given  were  erased,  and  the  lines  show  the  posi-  3- 
tions  of  the  missing  figures;  find  the  missing       36 

figures.  l  ~ 

6  — 3  — 

275.  What  is  the  side  of  a  square  field  which  contains 
3  A.  96sq.rd.? 

276.  A  thin  rectangular  lamina  of  metal,  3  ft.  2  in. 
long  and  2  ft.  9  in.  wide,  has  cut  from  its  four  cor- 
ners four  squares  whose  sides  are  3  in.  long.  The  four 
projecting  portions  are  turned  up  at  right  angles  to 
the  rest  of  the  lamina,  and  thus  form  a  lidless  box.  Find 
the  capacity  of  the  box. 

277.  Perform  the  last  example  after  substituting  dm 
for  ft.  and  cm  for  in.  and  find  out  the  weight  of  pure 
water  the  box  would  hold;  answer  in  lb.,  etc.,  to  the 
nearest  gr. 


Art.  283.]  EXAMPLES.  329 

278.  An  accommodation  train  going  at  the  rate  of  25 
mi.  an  hr.  starts  on  a  journey  an  hour  before  an  ex- 
press train  which  goes  at  the  rate  of  40  mi.  an  hr.  The 
accommodation  train  arrives  15  min.  before  the  express. 
Find  the  length  of  the  journey  in  Km. 

279.  On  a  map  on  the  scale  of  6  in.  to  a  mi.  a  rec- 
tangular field  is  represented  by  a  space  1  in.  long  and 
\  in.  broad ;  find  its  area  in  A.  Also  find  how  many  yd. 
less  of  paling  would  be  required  to  enclose  a  square  field 
of  the  same  area. 

280.  A  cubical  block  of  metal  of  7.84  in.  thick  weighs 
.25  lb.  per  cu.  in.  A  hole  of  square  sectional  area  is  to  be 
cut  completely  through  the  metal,  perpendicular  to  a  face 
of  the  cube,  in  order  that  the  weight  of  metal  left  may 
be  100  lb.  Find,  to  three  places  of  decimals,  the  side 
of  the  square  section. 


OF  THE     "^ 

UNIVERSITY 


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The  Algebras  by  Messrs.  Hall  and  Knight  have  been  introduced  in 
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A  TREATISE   ON   ALGEBRA. 

By  CHARLES  SMITH,  M.A. 

Cloth.     $1.90. 


No  stronger  commendation  of  this  work  is  needed  than  the  fact  that  it  is  the  text 
used  in  a  large  number,  if  not  in  the  majority,  of  the  leading  colleges  of  the  country, 
among  which  may  be  mentioned  Harvard  University,  Cornell  University,  University 
of  Ohio,  of  Pennsylvania,  of  Michigan,  of  Wisconsin,  of  Kansas,  of  California,  of 
Missouri,  Stanford  University,  etc.,  etc. 

"  Those  acquainted  with  Mr.  Smith's  text-books  on  conic  sections  and  solid 
geometry  will  form  a  high  expectation  of  this  work,  and  we  do  not  think  they  will 
be  disappointed.  Its  style  is  clear  and  neat,  it  gives  alternative  proofs  of  most  of 
the  fundamental  theorems,  and  abounds  in  practical  hints,  among  which  we  may 
notice  those  on  the  resolution  of  expressions  into  factors  and  the  recognition  of  a 
series  as  a  binominal  expansion."  —  Oxford  Review. 


HIGHER  ALGEBRA  FOR  SCHOOLS. 

By  H.  S.  HALL,  B.A.,  and  S.  R.  KNIGHT,  B.A. 


Cloth.     $1.90. 


"The  'Elementary  Algebra,'  by  the  same  authors,  which  has  already  reached  a 
sixth  edition,  is  a  work  of  such  exceptional  merit  that  those  acquainted  with  it  will 
form  high  expectations  of  the  sequel  to  it  now  issued.  Nor  will  they  be  disappointed. 
Of  the  authors'  '  Higher  Algebra,'  as  of  their  '  Elementary  Algebra,'  we  un- 
hesitatingly assert  that  it  is  by  far  the  best  work  of  the  kind  with  which  we  are 
acquainted.    It  supplies  a  want  much  felt  by  teachers."  —  The  Athenceum. 


MACMILLAN   &   CO., 

66  FIFTH  AVENUE,   NEW  YORK. 


ELEMENTARY  TRIGONOMETRY 


H.  S.  HALL,  B.A.,  and  S.  R.  KNIGHT,  B.A. 

Authors  of  " Algebra  for  Beginners"  "Elementary  Algebra  for  Schools,"  etc. 
Cloth.    $1.10. 

"  I  consider  the  work  as  a  remarkably  clean  and  clear  presentation  of  the  principles 
of  Plane  Trigonometry.  For  the  beginner,  it  is  a  book  that  will  lead  him  step  by  step 
to  grasp  its  subject  matter  in  a  most  satisfactory  manner."  —  E.  Miller,  University 
of  Kansas. 

"  The  book  is  an  excellent  one.  The  treatment  of  the  fundamental  relations  of 
angles  and  their  functions  is  clear  and  easy,  the  arrangement  of  the  topics  such  as 
cannot  but  commend  itself  to  the  experienced  teacher.  It  is,  more  than  any  other 
work  on  the  subject  that  I  just  now  recall,  one  which  should,  I  think,  give  pleasure 
to  the  student."  —  John  J.  Schobinger,  The  Harvard  School. 


WORKS  BY  REV.  J.  B.  LOCK. 

TRIGONOMETRY  FOR  BEGINNERS. 

AS  FAR  AS  THE  SOLUTION  OF  TRIANGLES. 

1  6mo.    75  cents. 

"A  very  concise  and  complete  little  treatise  on  this  somewhat  difficult  subject  for 
boys;  not  too  childishly  simple  in  its  explanations;  an  incentive  to  thinking,  not  a 
substitute  for  it.  The  schoolboy  is  encouraged,  not  insulted.  The  illustrations  are 
clear.  Abundant  examples  are  given  at  every  stage,  with  answers  at  the  end  of  the 
book,  the  general  correctness  of  which  we  have  taken  pains  to  prove.  The  definitions 
are  good,  the  arrangement  of  the  work  clear  and  easy,  the  book  itself  well  printed." 
—  Journal  of  Education. 


ELEMENTARY  TRIGONOMETRY. 

6th  edition.     (In  this  edition  the  chapter  on  Logarithms  has  been  carefully  revised.) 
16mo.    $1.10. 

"  The  work  contains  a  very  large  collection  of  good  (and  not  too  hard)  examples. 
Mr.  Lock  is  to  be  congratulated,  when  so  many  Trigonometries  are  in  the  field,  on 
having  produced  so  good  a  book ;  for  he  has  not  merely  availed  himself  of  the  labors 
of  his  predecessors,  but  by  the  treatment  of  a  well-worn  subject  has  invested  the 
study  of  it  with  interest."  —  Nature. 


MACMILLAN  &  CO., 

66   FIFTH  AVENUE,  NEW  YORK. 


UNIVERSITY  OF   C 

THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
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THIS    BOOK   ON    THE    DATE   DUE.    THE   PENALTY 
WILL  INCREASE  TO  50  CENTS  ON  THE  FOURTH 
DAY    AND    TO     $1.00     ON     THE    SEVENTH     DAY 
OVERDUE. 

APR     2  1035 

DEC    7  1935 

LD  21-100m-8,'34 

._ 


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